\newconstantfamily

Ctildesymbol=~C

Finite Time Hyperbolic Coordinates

Stefano Luzzatto Dominic Veconi Khadim War

Abstract

We define finite-time hyperbolic coordinates, describe their geometry, and prove various results on both their convergence as the time scale increases, and on their variation in the state space. Hyperbolic coordinates reframe the classical paradigm of hyperbolicity: rather than define a hyperbolic dynamical system in terms of a splitting of the tangent space into stable and unstable subspaces, we define hyperbolicity in terms of the co-eccentricity of the map. The co-eccentricity describes the distortion of unit circles in the tangent space under the differential of the map. Finite-time hyperbolic coordinates have been used to demonstrate the existence of SRB measures for the Hénon map; our eventual goal is to both elucidate these techniques and to extend them to a broad class of nonuniformly and singular hyperbolic systems.

1 Introduction

1.1 Physical Measures

One of the most interesting and challenging problems in the theory of dynamical systems is that of describing the statistical properties of a map $f:M\to M$ on some (typically compact) metric space $M$ . Given an initial condition $x\in M$ , for every $n\geq 1$ , we can define the empirical measure

e_{n}(x):=\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^{i}(x)}

associated to the orbit of $x$ . Notice that $e_{n}(x)$ is an atomic probability measure uniformly distributed on the first $n$ points of the orbit of $x$ . The measure $e_{n}$ describes the frequency with which the orbit visits various regions in $M$ . If the sequence of empirical measures converges in the weak topology to a probability measure $\mu$ as $n\to\infty$ , then $\mu$ can be thought of as describing the asymptotic statistical distribution of the orbit. In this situation, we say that the point $x$ has statistical behavior described by the probability distribution $\mu$ . We define the basin of $\mu$ as the set $B_{\mu}$ of points whose asymptotic statistical distribution is described by the measure $\mu$ . Formally,

\mathcal{B}_{\mu}:=\{x\in M:e_{n}(x)\to\mu\}.

If $M$ is endowed with a normalized reference probability measure $m$ (often a Lebesgue measure or Riemannian area form), we say that $\mu$ is a physical measure if

m(\mathcal{B}_{\mu})>0

since this means that there is a “physically observable” set of points whose asymptotic statistical distribution is described by the given measure $\mu$ . If there is some finite set of probability measures such that the union of the corresponding basins has full Lebesgue measure, then we can say in some sense that we have succeeded in describing the dynamics from a probabilistic/statistical point of view. A major and ongoing line of research over the last several decadesis establishing the existence and uniqueness of physical measures in specific classes of smooth dynamical systems.

Before we discuss in the forthcoming section some technqiues for constructing physical measures, we first remark that the existence of physical measures is not guaranteed: there are dynamical systems that admit no physical measures at all. The simplest example is the identity map $f(x)=x$ , for which $e_{n}(x)=\delta_{x}$ for every $x$ and $n\geq 1$ , and therefore trivially $e_{n}(x)\to\delta_{x}$ . It follows that there cannot be any physical measures, since $\mathcal{B}_{\delta_{x}}=\{x\}$ , and therefore $m(B_{\delta_{x}})=0$ for any Lebesgue reference measure $m$ since a Lebesgue reference measure must be nonatomic. More interesting examples are maps for which the empirical measures $e_{n}(x)$ do not converge. This can be very counter-intuitive since if there are two distinct probability measures $\mu,\nu$ and subsequences $n_{i},n_{j}\to\infty$ such that $e_{n_{i}}(x)\to\mu$ and $e_{n_{j}}(x)\to\nu$ , this means that the statistical distribution of the orbit depends on the time scale. In a rough sense, this means that the statistical behavior of the map is sometimes described by the measure $\mu$ and sometimes by the measure $\nu$ . This means that there is no well defined asymptotic statistical distribution, in which case we say that the orbit of $x$ exhibits non-statistical, or historic, behaviour, see [10, 13, 14, 20, 21]. The Palis Conjecture [17] says that non-statistical dynamical systems are “exceptional,” whereas “typical” dynamical systems admit a finite number of physical measures whose unions of basins have full Lebesgue measure.

1.2 Hyperbolicity

The simplest examples of physical measures are Dirac-delta measures on attracting periodic orbits. However, there are huge classes of systems which do not have attracting periodic orbits and for which proving the existence of physical measures is highly challenging. Over the last 50 years, starting mostly from the work of Sinai, Bowen, and Ruelle in the 1970s [6, 7, 19], there has been a huge amount of research on developing techniques to prove the existence of physical measures. Most of these techniques assume some kind of hyperbolicity, which essentially consists of a $Df$ -invariant tangent bundle decomposition

T_{x}M=E^{1}_{x}\oplus\cdots\oplus E^{\kappa}_{x}

(1)

satisfying various properties. These properties include estimates on the contraction or expansion of vectors in each sub-bundle, relation between the contraction and expansion in different sub-bundles, and the regularity of the decomposition (i.e., whether the distributions $E^{j}_{x}$ depend measurably or continuously on $x$ ). See [2] for a comprehensive survey. While there is no completely general result which says that every form of hyperbolicity implies the existence of a physical measure, there are many highly non-trivial results which show that even some very weak hyperbolicity conditions can be sufficient to prove the existence of a physical measure [8].

1.3 Verifying Hyperbolicity

Notwithstanding the importance of the results mentioned above, using hyperbolicity as an assumption naturally leads to the question of whether it can be verified in specific systems. As it turns out, there are many situations in which hyperbolicity can be verified relatively easily (at least in principle), either through geometric and analytic arguments [18] and/or explicit rigorous numerical computations [1]. Most of these easily verifiable situations occur in the context of“uniformly hyperbolic” systems, in which the hyperbolic splitting is continuous, robust under small perturbations, and the contraction and expansion are uniform in the phase space.

However, there are more general situations with broader applications than uniform hyperbolicity, in which the hyperbolic splitting is only measurable, the splitting can be easily destroyed under small perturbations, and the contraction and expansion estimates cannot be given uniformly in the phase space. Here we loosely refer to these dynamical systems as “nonuniformly hyperbolic.” The verification of the existence of physical measures in nonuniformly hyperbolic systems is still extremely hard. A major breakthrough was made in 1991 by Benedicks and Carleson in their famous paper [3], in which they studied the Hénon family $H_{a,b}(x,y)=(1+y-ax^{2},bx)$ of two-dimensional diffeomorphisms of $\mathbb{R}^{2}$ . They developed a number of very delicate and sophisticated arguments and estimates, which we will for simplicity refer to as parameter-exclusion arguments, to prove that when the map is sufficiently strongly dissipative (i.e. when the parameter $b$ is sufficiently small), there exists a positive Lebesgue measure set of parameters $a$ for which the map $H_{a,b}$ admits some (nonuniformly) hyperbolic structure. It was then shown in [4] that this hyperbolic structure implies the existence of a physical measure.

There exist several extensions and generalizations of the arguments and results of Benedicks and Carleson [3], but they typically deal with families of maps that share a lot of the key features of the Hénon maps, such as smoothness and strong dissipativity. There are many classes of nonuniformly hyperbolic dynamical systems that do not share these features, such as the famous and very well-studied Standard Map [5, 9], which is area-preserving and therefore not dissipative. A family of dissipative examples of nonuniformly hyperbolic systems come from the Lorenz equations [15], whose two-dimensional Poincaré “Lorenz” map lacks smoothness: the Poincaré map contains a non-trivial discontinuity curve, giving rise to a curve of singularities with unbounded derivatives. For certain parameter values, non-rigorous numerical studies strongly suggest the existence of some kind of hyperbolic structure, but the rigorous verification of this remains an open problem and the existing techniques do not seem to easily generalize to these systems.

1.4 Goals of this paper

The original pioneering parameter-exclusion arguments of Benedicks and Carleson, as well as those developed in its generalizations, are extremely intricate, consisting of many “sub-arguments” that are tightly inter-woven. We believe that this is the main reason for which only a handful of researchers have taken the time to properly understand the methodology of the proof, and the reason for which there have been no significant generalizations to this methodology in the literature outside of strongly dissipative smooth settings.

This is the first in a series of papers whose goal is to gradually disentangle the various strands of the existing parameter-exclusion arguments. Our hope is to not only make the original argument much more accessible, but also highlight that many of the sub-arguments in the construction are of independent interest and can be generalized to maps which are not necessarily strongly dissipative and may have discontinuities or singularities. Ultimately this should make it possible to develop generalizations of the entire parameter-exclusion arguments to systems such as the Standard map or Lorenz-like maps.

We begin this program with a careful study of a comparatively elementary notion, referred to in [3] as the most contracted direction of the differential map. This is somewhat understated in the original arguments, but turns out to play a crucial role in constructing some geometric “hyperbolic” structures that depend on only finitely many iterates of the map. By comparison, the usual notion of hyperbolicity is essentially asymptotic and therefore requires information about all iterates. Various properties of these finite-time most contracted directions, such as their dependence on the iterations and the base point, constitute important conditions for the development of the overall argument. These properties seem to rely significantly on the smoothness and especially the strong dissipativity of the maps.

In this paper we will give a formal definition of what we call (finite time) Hyperbolic Coordinates, which we believe is the natural and more intuitive setting in which to understand and study the “most contracted directions” of [3]. We will introduce a general and quite weak pointwise and finite-time hyperbolicity condition, which we call quasi-hyperbolicity, and show that under this assumption the hyperbolic coordinates satisfy a number of important properties. These properties, in particular, include those required for the parameter-exclusion arguments of [3] and its generalizations. Crucially, however, our quasi-hyperbolicity condition does not require the map to be strongly dissipative: it can apply even to area-preserving maps such as the Standard map. Moreover, nor does our condition require the map to have a bounded derivative, thus making it applicable to systems with singularities such as Lorenz-like maps. This broad applicability makes the results completely new as they are not included, even implicitly, in any of the existing literature as far as the authors are aware.

2 Definitions and Statements of Results

In Section 2.1 we give the definition of finite-time hyperbolic coordinates and make several remarks about the motivation for this notion. Then in Section 2.2 we introduce our condition of pointwise and finite time quasi-hyperbolicity, and in Section 2.3 we state our main results.

2.1 Hyperbolic Coordinates

In this section we give the formal definition of Hyperbolic Coordinates and discuss some of their properties. We assume that $M$ is a Riemann surface and that $\Phi:M\to M$ is a map. Our results are pointwise in the sense that they apply to individual orbits, so we do not assume any global regularity of $\Phi$ . Instead, we fix some $\xi_{0}\in M$ and some $k\geq 1$ and suppose $\Phi^{k}$ is $C^{2}$ at $\xi_{0}$ (for the definition we only need $C^{1}$ but for many properties and for our results we will need $C^{2}$ ).

2.1.1 Definition of Co-eccentricity and Hyperbolic Coordinates

We define the co-eccentricity of $\Phi^{k}_{\xi_{0}}$ or, more precisely, of the derivative map $D\Phi^{k}_{\xi_{0}}:T_{\xi_{0}}M\to T_{\xi_{k}}M$ , as

C_{\xi_{0},k}:=\frac{|\det(D\Phi^{k}_{\xi_{0}})|}{\|D\Phi_{\xi_{0}}^{k}\|^{2}}=\frac{\|(D\Phi_{\xi_{0}}^{k})^{-1}\|^{-2}}{|\det(D\Phi^{k}_{\xi_{0}})|}=\frac{\|(D\Phi_{\xi_{0}}^{k})^{-1}\|^{-1}}{\|D\Phi_{\xi_{0}}^{k}\|}.

(2)

Notice that $\|(D\Phi_{\xi_{0}}^{k})^{-1}\|^{-1}$ is a somewhat convoluted way of writing the co-norm of $D\Phi_{\xi_{0}}^{k}$ , that is, the norm of the image of the most contracted unit vector. The equality between the three expressions in (2) then follows immediately from the fact that $\det(D\Phi^{k}_{\xi_{0}})=\|D\Phi_{\xi_{0}}^{k}\|\|(D\Phi_{\xi_{0}}^{k})^{-1}\|^{-1}$ . The third formulation in (2) clearly shows that we always have $C_{\xi_{0},k}\leq 1$ . Letting $\mathcal{S}_{0}\subset T_{\xi_{0}}M$ be the unit circle and $\mathcal{S}_{k}:=D\Phi_{\xi_{0}}^{k}(\mathcal{S}_{0})\subset T_{\xi_{k}}M$ be its image, the co-eccentricity has a very natural geometrical interpretation: if $C_{\xi_{0},k}=1$ , then $\mathcal{S}_{k}$ is also the unit circle, whereas if $C_{\xi_{0},k}<1$ , then $\mathcal{S}_{k}$ is a non-trivial ellipse and there are distinct unit vectors $e^{(k)},f^{(k)}\in T_{\xi_{0}}M$ that map to the minor and major semi-axes of the ellipse $\mathcal{S}_{k}$ , and are therefore respectively the most contracted and most expanded unit vectors for $D\Phi^{k}_{\xi_{0}}$ .

Definition 2.1.

If $C_{\xi_{0},k}<1$ , the coordinates

\mathcal{H}^{(k)}=\{e^{(k)},f^{(k)}\}

defined by taking $e^{(k)},f^{(k)}$ as unit basis vectors, are called hyperbolic coordinates of order $k$ at $x_{0}$ .

Notice that $e^{(k)}$ is the most contracted unit vector and $f^{(k)}$ as the most expanded unit vector under $D\Phi_{\xi_{0}}^{k}$ , but these are just relative terms and these vectors may not actually be expanded or contracted at all. Notice also that hyperbolic coordinates are not uniquely defined since $-e^{(k)}$ and $-f^{(k)}$ are also most contracted and most expanded respectively. We therefore just assume that some choice has been made and, as we shall see, this will not create any ambiguity or confusion in the settings which we will consider.

2.1.2 Hyperbolic Coordinates as Diagonalizing Coordinates

Hyperbolic coordinates are useful in a number of ways. First of all, note that they form an orthonormal basis of $T_{\xi_{0}}M$ : if $\mathcal{S}_{0}\subset T_{\xi_{0}}M$ is the unit circle and $\mathcal{S}_{k}=D\Phi_{\xi_{0}}^{k}(\mathcal{S}_{0})\subset T_{\xi_{k}}M$ is the ellipse given by the image of $\mathcal{S}_{0}$ under $D\Phi_{\xi_{0}}^{k}$ , then it is a fundamental result in linear algebra that the minor and major axes of $\mathcal{S}_{k}$ have orthogonal preimages in $\mathcal{S}_{0}$ (see Remark 2.1 below), and these preimages are precisely $e^{(k)}$ and $f^{(k)}$ . For any $i\geq 1$ , we let

e^{(k)}_{i}:=D\Phi^{i}(e^{(k)})\quad\text{ and }\quad f^{(k)}_{i}:=D\Phi^{i}(f^{(k)}).

(3)

Notice that $e^{(k)}_{i},f^{(k)}_{i}\in T_{\xi_{i}}M$ , where $\xi_{i}=\Phi^{i}(\xi_{0})$ , and $e^{(k)}_{k}$ and $f^{(k)}_{k}$ are by definition minor and major semi-axes of the ellipse $\mathcal{S}_{k}$ and so are also orthogonal (which is not generally the case when $i\neq k$ ). Normalizing these vectors we can define an orthonormal basis in $T_{\xi_{k}}M$ given by the unit vectors

\mathcal{H}^{(k)}_{k}:=\{{e^{(k)}_{k}}/{\|e^{(k)}_{k}\|},{f^{(k)}_{k}}/{\|f^{(k)}_{k}\|}\}.

In coordinates $\mathcal{H}^{(k)}$ in $T_{\xi_{0}}M$ and $\mathcal{H}^{(k)}_{k}$ in $T_{\xi_{k}}M$ , the derivative $D\Phi^{k}_{\xi_{0}}:T_{\xi_{0}}M\to T_{\xi_{k}}M$ has diagonal form

D\Phi^{k}_{\xi_{0}}=\begin{pmatrix}\|f^{(k)}_{k}\|&0\\ 0&\|e^{(k)}_{k}\|\end{pmatrix}=\begin{pmatrix}\|D\Phi^{k}_{\xi_{0}}\|&0\\ 0&\|(D\Phi^{k}_{\xi_{0}})^{-1}\|^{-1}\end{pmatrix}.

(4)

This diagonal form of the derivative can be very useful in a number of situations.

Remark 2.1.

In view of (4), hyperbolic coordinates in smooth dynamics correspond to the singular value decomposition of the linear operators $D\Phi_{\xi_{0}}^{i}:T_{\xi_{0}}M\to T_{\Phi^{i}(\xi_{0})}M$ . In general, a linear map $A:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is expressible as $A=U\Sigma V^{*}$ , where $U\in\mathrm{O}(m),V\in\mathrm{O}(n)$ are orthogonal matrices and $\Sigma$ is an $m\times n$ matrix whose non-diagonal entries are all $0$ . The eigenvalues of $A^{*}A$ are the squares of the diagonal entries of $\Sigma$ (the “singular values”), where $A^{*}$ is the adjoint of $A$ , and the corresponding eigenvectors are the columns of $V$ . In $\mathbb{R}^{2}$ , these eigenvectors are the directions of maximal and minimal expansion. So, $e^{(k)}$ and $f^{(k)}$ are eigenvectors of the matrix $(D\Phi^{k}_{\xi_{0}})^{*}\circ D\Phi_{\xi_{0}}^{k}$ , where $(D\Phi_{\xi_{0}}^{k})^{*}:T_{\Phi^{k}(\xi_{0})}M\to T_{\xi_{0}}M$ is the adjoint of $D\Phi_{\xi_{0}}^{k}$ with respect to the Riemannian inner product in $T_{\xi_{0}}M$ and $T_{\Phi^{k}(\xi_{0})}M$ , and these eigenvectors have corresponding eigenvalues $\|e^{(k)}_{k}\|^{2}$ and $\|f^{(k)}_{k}\|^{2}$ .

2.1.3 Finite-time stable and unstable manifolds

A second important observation is that we can extend hyperbolic coordinates to a neighbourhood of the base point $\xi_{0}$ since, if $C_{\xi_{0},k}<1$ at $\xi_{0}$ then, since $\Phi^{k}$ is assumed to be $C^{1}$ , the same will be true in a neighbourhood of $\xi_{0}$ . There exist therefore in this neighbourhood two orthogonal unit vector fields $e^{(k)},f^{(k)}$ given by the most contracted and most expanded direction at each point. Moreover, hyperbolic coordinates can be computed explicitly in terms of the partial derivatives of $D\Phi^{k}$ : parametrizing the unit circle by $\mathcal{S}=\{(\sin\theta,\cos\theta),\theta\in[0,2\pi)\}$ , the angles which map to the minor and major axes of the ellipse $\mathcal{S}_{k}$ are solutions to the equation ${d}\|D\Phi^{k}(\sin\theta,\cos\theta)\|/{d\theta}=0,$ which gives

\tan 2\theta=\frac{2(\partial_{x}\Phi_{1}^{k}\partial_{y}\Phi_{1}^{k}+\partial_{x}\Phi_{2}^{k}\partial_{y}\Phi_{2}^{k})}{(\partial_{x}\Phi_{1}^{k})^{2}+(\partial_{x}\Phi_{2}^{k})^{2}-(\partial_{y}\Phi_{1}^{k})^{2}-(\partial_{y}\Phi_{2}^{k})^{2}}.

(5)

This gives an alternative proof of the fact that $e^{(k)},f^{(k)}$ are orthogonal and also shows that they depend on the base point with the same regularity as the partial derivatives of $\Phi^{k}$ . In particular, if $\Phi^{k}$ is $C^{2}$ in a neighbourhood of $\xi_{0}$ then the unit vectors $e^{(k)},f^{(k)}$ define two orthogonal $C^{1}$ vector fields and are therefore locally integrable and define two orthogonal foliations $\mathcal{E}^{(k)}$ , $\mathcal{F}^{(k)}$ . The leaves of these foliations are the integral curves of the most contracted and most expanded directions for $D\Phi^{k}$ and therefore can naturally be thought of as (finite time) stable and unstable manifolds (of order $k$ ). This idea has been developed in [11, 12] to give new proofs of the classical stable manifold theorems in certain two-dimensional settings, including for orbits which exhibit very weak forms of hyperbolicity.

Extending the notation introduced in (3) above, we let $\mathcal{E}^{(k)}_{i}:=\Phi^{i}(\mathcal{E}^{(k)})$ and $\mathcal{F}^{(k)}_{i}:=\Phi^{i}(\mathcal{F}^{(k)})$ denote the images of these stable and unstable foliations, which are themselves the foliations given by the integral curves of the vector fields $e^{(k)}_{i},f^{(k)}_{i}$ . In particular, the foliations $\mathcal{E}^{(k)}_{k},\mathcal{F}^{(k)}_{k}$ are orthogonal and therefore we can use the diagonal form of the derivative given in (4) in a neighbourhood of the point $\xi_{0}$ .

2.2 Quasi-hyperbolicity

As we have seen in the previous section, hyperbolic coordinates give rise to some dynamically significant geometric structures, in particular the orthogonal foliations in which the derivative has the especially simple diagonal form (4). However, the usefulness of the coordinates depends on how much information we have about these foliations, such as the direction of the leaves and their curvatures. In principle, a lot of information can be obtained from the formula in (5), but in practice this can really be used only for the first iterate $k=1$ , as we do not generally have enough explicit information about the partial derivatives for higher iterates. We therefore need to take a different approach which uses somewhat “coarser” information about the derivative along the orbit, but is still sufficient, in some cases, to deduce relevant bounds for the geometry of the hyperbolic coordinates and the corresponding foliations.

We formulate a notion of $\mathfrak{C}$ -quasi-hyperbolicity along the orbit of a point in terms of a set $\mathfrak{C}$ of constants. The conditions involved in this formulation may appear at first sight somewhat technical, but are in fact quite natural and quite mild. While our goal is to formulate this notion for singular systems with unbounded derivative, our results are also highly relevant in the simpler setting of non-singular systems in which the derivative is uniformly bounded. In the non-singular situation, the formulation is a bit simpler, so for the sake of clarity, we formulate our definitions in the non-singular setting first.

2.2.1 Quasi-hyperbolicity in non-singular systems

Definition 2.2.

Given a set $\mathfrak{C}=\{\Gamma,\lambda,b,c\}$ of positive constants, the point $\xi_{0}$ is $\mathfrak{C}$ -quasi-hyperbolic at time $k$ if there exists constants $C>0,B,D\geq 1$ such that for every $1\leq i\leq k$ the map $\Phi^{i}$ is $C^{2}$ at $\xi_{0}$ and satisfies

C\lambda^{i}<\|D\Phi^{i}_{\xi_{0}}\|<D\Gamma^{i}\quad\text{ and }\quad C_{\xi_{0},i}:=\frac{\|(D\Phi_{\xi_{0}}^{i})^{-1}\|^{-1}}{\|D\Phi_{\xi_{0}}^{i}\|}<Bc^{i}<1,

(6)

and

\|D\Phi_{\xi_{i-1}}\|,\|D^{2}\Phi_{\xi_{i-1}}\|<D\Gamma\quad\text{ and }\quad\det D\Phi_{\xi_{i-1}}\leq b

(7)

with the constants satisfying

\Gamma\geq\max\{\lambda,1\},\quad\quad b<\lambda^{2},\quad\quad c<\lambda^{2}/\Gamma^{2}<1.

(8)

We make several remarks about the interpretation and significance of these conditions before stating their generalization to the singular setting.

Remark 2.2.

The bounds in (6) are the two core “hyperbolicity” conditions, albeit, and crucially, formulated in a way that does not require an a priori decomposition of the tangent bundle.

Remark 2.3.

The lower and upper bounds on $\|D\Phi^{i}_{\xi_{0}}\|$ in (6) are in some sense “trivial” since such bounds always exist, but the purpose here is to give these bounds in terms of specific constants which appear also in the other conditions (notice that they imply in particular a minimum “growth” of the norm of the derivative but we only assume $\lambda>0$ , not necessarily $\lambda>1$ , so this may not necessarily require actual growth). The ratio $\lambda/\Gamma$ , which is always $\leq 1$ , may in some situations be chosen very close to 1. For instance, suppose that $\xi_{0}$ is a typical point for an invariant probability measure $\mu$ , “typical” in the sense that the Lyapunov exponent $\chi=\lim_{n\to\infty}n^{-1}\log\|D\Phi^{i}_{\xi_{0}}\|$ is well defined. This implies that for any $\epsilon>0$ and suitable constants $C$ and $D$ (depending on $\epsilon$ ), the bounds on $\|D\Phi^{i}_{\xi_{0}}\|$ in (6) are satisfied with $\lambda=e^{\chi-\epsilon}$ and $\Gamma=e^{\chi+\epsilon}$ . Then $\lambda/\Gamma=e^{-2\epsilon}$ , which can be made arbitrarily close to 1 by taking $\epsilon$ small.

Remark 2.4.

We also emphasize that we do not assume $\lambda>1$ , and the second set of inequalities in (6) essentially say there is is also a “contracting” direction, albeit just contracting relative to some “more expanding” direction (which may not even be expanding). This is thus essentially a weak “dominated decomposition” condition. Notice that the bound is formulated in terms of the constant $c$ which is bounded above by the ratio $\lambda^{2}/\Gamma^{2}$ . This puts some restrictions on its range of applicability but, as mentioned in Remark 2.3, there are many cases in which $\lambda$ and $\Gamma$ can be chosen so that $\lambda/\Gamma$ is very close to 1, allowing this condition to be quite easily satisfied.

Remark 2.5.

The conditions in (6) and (7) could morally be stated directly in terms of $\lambda$ and $\Gamma$ , without reference to the constants $b$ and $c$ , but for technical reasons we require some uniform bounds independent of $k$ which are achieved by introducing the constants $b,c$ , which can be thought of as “arbitrarily close” to $\lambda^{2}$ and $\lambda^{2}/\Gamma^{2}$ respectively.

Remark 2.6.

The non-singularity of the map is reflected in the uniform upper bound for the norms of the first and second derivatives in the first expression in (7). We will have to relax this in the general setting.

Remark 2.7.

The determinant is not required to be small. In many cases we have $\lambda>1$ and therefore the bound (7) on $b$ is not very restrictive at all, allowing us to apply our results even to area-preserving systems.

Remark 2.8.

Strictly speaking the set of constants $\mathfrak{C}$ which define quasi-hyperbolicity also includes the constants $B,C,D$ . These latter constants will come into the definition of some constants which appear in our results, but there are no restrictions on them for the definition of quasi-hyperbolicity. Therefore, for clarity, we have not included them in the “core” constants $\mathfrak{C}$ .

Remark 2.9.

The assumption that $B,D\geq 1$ (as opposed to simply $B,D>0$ ) is an assumption based on technical convenience. Since $B$ and $D$ are used in upper bounds, we lose no generality in assuming $B,D\geq 1$ . This also holds for $B$ and $D$ in Definition 2.3 below.

2.2.2 Quasi-hyerbolicity in singular systems

We now generalize the definition above to singular systems in which the derivative may be unbounded.

Definition 2.3.

[(Singular) Quasi-Hyperbolicity] Given a set $\mathfrak{C}=\{\Gamma,\widetilde{\Gamma},\lambda,b,c,\tilde{c}\}$ of positive constants, the point $\xi_{0}$ is $\mathfrak{C}$ -quasi-hyperbolic at time $k$ if there exists constants $B,D\geq 1\geq\widetilde{B},C>0$ such that for every $1\leq i\leq k$ the map $\Phi^{i}$ is $C^{2}$ at $\xi_{0}$ and

C\lambda^{i}<\|D\Phi^{i}_{\xi_{0}}\|<D\Gamma^{i}\quad\text{ and }\quad C_{\xi_{0},i}:=\frac{\|(D\Phi_{\xi_{0}}^{i})^{-1}\|^{-1}}{\|D\Phi^{i}\|}\leq Bc^{i}<1,

(9)

and

\|D\Phi_{\xi_{i-1}}\|,\|D^{2}\Phi_{\xi_{i-1}}\|<D\Gamma\widetilde{\Gamma}^{i-1},\quad\text{ and }\quad\det D\Phi_{\xi_{i-1}}\leq b.

(10)

We assume moreover that

\widetilde{\Gamma}\geq 1\quad\text{ and }\quad\Gamma>\max\{\lambda,1\}\quad\text{ and }\quad b<\Gamma^{2}\widetilde{\Gamma}

(11)

and either

b<\lambda^{2}/\widetilde{\Gamma}\quad\text{ and }\quad c<\lambda^{3}/\Gamma^{3}\widetilde{\Gamma}^{3}<1,

(I)

in which case we say that $\xi_{0}$ is $\mathfrak{C}$ -quasi-hyperbolic of type (I), and/or

C_{\xi_{i-1},1}:=\frac{\|(D\Phi_{\xi_{i-1}})^{-1}\|^{-1}}{\|D\Phi_{\xi_{i-1}}\|}\geq\widetilde{B}\tilde{c}^{i-1}\quad\text{ and }\quad b<\lambda^{2}\tilde{c}\quad\text{ and }\quad c<\lambda^{2}\tilde{c}^{2}/\Gamma^{2}\widetilde{\Gamma}<\tilde{c}\leq 1

(II)

in which case we say that $\xi_{0}$ is $\mathfrak{C}$ -quasi-hyperbolic of type (II).

We conclude this section with a number of additional remarks concerning our assumptions in the singular setting. These remarks are not formally needed for the statement of our results in Section 2.3, but are included to help clarify how the assumptions should be interpreted heuristically.

Remark 2.10.

The two core sets of conditions (9) and (10) are exactly identical to the conditions (6) and (7) respectively in the non-singular case except for the addition of the new constant $\widetilde{\Gamma}$ in (10), which now allows the derivative to be unbounded along the orbit, albeit in a controlled way. This is a significant generalization of the definition and hugely increases the range of systems to which it is applicable. The conditions on the constants in (11) and (I) are also very similar to the corresponding conditions (8) in the non-singular setting albeit incorporating the new constant $\widetilde{\Gamma}$ . The alternative condition (II) is not just a formal condition on the constants but introduces a requirement of a specific lower bound on the pointwise co-eccentricity $C_{\xi_{i},1}$ .

Remark 2.11.

The distinction between type (I) and type (II) singular hyperbolicity is not particularly relevant from a conceptual point of view. It is rather just a technical distinction motivated by the fact that we can address both situations by estimating some expressions in slightly different ways in the course of the proof, and one or the other might be easier to verify in some specific examples. The results we obtain are the same: they do not distinguish between these two cases except in the specific values of some of the constants.

Remark 2.12.

The bound on the one-step co-eccentricity in (II) is essentially just a mild bounded recurrence condition for the orbit near the singularity, as are also the pointwise bounds in (10). The constants $\tilde{c}$ and $\widetilde{\Gamma}$ are therefore related and we can even choose them satisfying an explicit relationship, such as $\tilde{c}=1/\widetilde{\Gamma}^{1/2}$ . Using the fact that $\Gamma>\lambda$ , this would imply $c<{\lambda^{2}\tilde{c}^{2}}/{\Gamma^{2}\widetilde{\Gamma}}<1/{\tilde{\Gamma}^{2}}=\tilde{c}^{4}\leq\tilde{c}\leq 1$ which shows that this choice is compatible with the last set of inequalities in (II).

Remark 2.13.

Taking $\tilde{c}=\widetilde{\Gamma}=1$ in (II) we recover exactly the conditions (8) of the non-singular setting of Definition 2.2 (the condition on the pointwise co-eccentricity and the constant $\widetilde{B}$ do not appear explicitly there but are automatically satisfied). We will therefore not give a separate proof of our results in the non-singular case since they are included as special cases of the singular case of type (II). Also, to simplify the terminology we will usually omit explicit reference to the set $\mathfrak{C}$ since this is understood to have been fixed.

Remark 2.14.

The constant $c$ is an upper bound of the the accumulated co-eccentricity of $D\Phi$ along the orbit $\xi_{i}$ , whereas $\tilde{c}$ is a lower bound of the one-step co-eccentricity. The assumption that $\tilde{c}>c$ is not contradictory with the fact that $c$ is an upper bound, while $\tilde{c}$ is a lower bound, for two reasons. Firstly, due to rotation effects, if $A_{1}$ and $A_{2}$ are two matrices with co-eccentricities $C_{A_{1}}$ and $C_{A_{2}}$ , then there is no relation between the product of the co-eccentricities $C_{A_{1}}C_{A_{2}}$ and the co-eccentricity of the product $C_{A_{1}A_{2}}$ . So there need be no relation between the accumulated co-eccentricity $C_{\xi_{0},i}$ and the one-step co-eccentricity $C_{\xi_{i},1}$ . Secondly, if there are no rotation effects, and the accumulated co-eccentricity is the product of the one-step co-eccentricities (as in the classical geometric Lorenz attractor), there is still no contradiction: since $C_{\xi_{0},i}\leq Bc^{i}$ and $C_{\xi_{i},1}\geq\widetilde{B}\tilde{c}^{i}$ , if $C_{\xi_{0},i}=\prod_{j=0}^{i-1}C_{\xi_{j},1}$ , we would have:

Bc^{i}\geq C_{\xi_{0},i}=\prod_{j=0}^{i-1}C_{\xi_{j},1}\geq\prod_{j=0}^{i-1}\widetilde{B}\tilde{c}^{j}=\widetilde{B}^{i}\tilde{c}^{i(i-1)/2}.

For appropriate choices of $B$ and $\widetilde{B}$ , it is therefore perfectly reasonable to suppose $\tilde{c}>c$ .

2.2.3 Auxiliary Constants

The statements of our main results below, as well as the intermediate computations in the argument, will involve a number of lengthy expressions involving the constants used in the definition of quasi-hyperbolicity. To simplify these expressions we will introduce a number of auxiliary constants at various steps of the proof. For ease of reference we collect the definitions of all these constants here. First of all, let

Q_{0}:=\sqrt{\frac{2}{1-B^{2}c^{2}}}\quad\text{ and }\quad K_{1}:=\frac{Q_{0}^{2}}{\sqrt{2}}

(12)

Notice that by (9), we have $Bc<1$ and therefore also $B^{2}c^{2}<1$ and so $Q_{0}$ and $K_{1}$ are well-defined positive constants. Moreover, assuming the constant $B$ is fixed, we have that $Q_{0},K_{1}\to\sqrt{2}$ as $c\to 0$ . Then we let

Q_{1}:=BD+\frac{Q_{0}BD^{3}\Gamma}{C(\lambda-\Gamma\widetilde{\Gamma}c)},\quad Q_{2}:=\frac{1}{C}+\frac{Q_{0}D^{2}\Gamma\lambda}{C^{2}(\lambda^{2}-\widetilde{\Gamma}b)},\quad Q_{3}:=\frac{Q_{1}D\Gamma^{2}\widetilde{\Gamma}}{\lambda},\quad Q_{4}:=\frac{Q_{1}Q_{2}D\Gamma^{5}\widetilde{\Gamma}^{4}}{\lambda^{2}(\lambda^{3}-\Gamma^{3}\widetilde{\Gamma}^{3}c)}.

These will be used in the setting of type (I) quasi-hyperbolicity. Then, by (11) we have $\lambda/\Gamma\widetilde{\Gamma}<1$ and therefore (I) gives $c<\lambda^{3}/\Gamma^{3}\widetilde{\Gamma}^{3}<\lambda/\Gamma\widetilde{\Gamma}<1$ which implies that $\lambda>\Gamma\widetilde{\Gamma}c$ and therefore the denominators in the definition of $Q_{1}$ and $Q_{4}$ are strictly positive. Similarly, from (I) we have $\lambda^{2}>\widetilde{\Gamma}b$ which implies that the denominator in the definition of $Q_{2}$ is strictly positive. It follows that under the assumptions (I), $Q_{1}$ - $Q_{4}$ are all well-defined positive constants. Moreover, since $Q_{0}\to\sqrt{2}$ as $c\to 0$ it follows that $Q_{1},Q_{2},Q_{4}$ are monotonic in $c$ and decrease to positive constants as $c\to 0$ , whereas $Q_{3}$ is independent of $c$ .

We now let

\widetilde{Q}_{1}:=BD+\frac{Q_{0}B\tilde{c}}{\widetilde{B}(\tilde{c}-c)},\quad\widetilde{Q}_{2}:=\frac{1}{C}+\frac{Q_{0}D\lambda^{2}\tilde{c}}{\widetilde{B}C^{2}(\lambda^{2}\tilde{c}-b)}.\quad\widetilde{Q}_{3}:=\widetilde{Q}_{1}D\Gamma,\quad\widetilde{Q}_{4}:=\frac{\widetilde{Q}_{1}\widetilde{Q}_{2}D\Gamma^{4}\widetilde{\Gamma}}{\lambda^{2}(\lambda^{2}\tilde{c}^{2}-\Gamma^{2}\widetilde{\Gamma}c)}.

These will be used in the setting of type (II) quasi-hyperbolicity, in which case it follows from (II) that $\widetilde{Q}_{1}$ - $\widetilde{Q}_{4}$ are are well-defined and positive. Moreover, $\widetilde{Q}_{1},\widetilde{Q}_{2},\widetilde{Q}_{4}$ are monotonic in $c$ and decrease to positive constants as $c\to 0$ , whereas $\widetilde{Q}_{3}$ is independent of $c$ . Finally we let

Q:=\frac{BD^{3}\Gamma^{4}\widetilde{\Gamma}}{C^{2}\lambda^{2}(\Gamma^{2}\widetilde{\Gamma}-b)}\quad\text{ and }\quad K_{2}:=\max\{K_{1}(Q_{3}+Q_{4}+Q),K_{1}(\widetilde{Q}_{3}+\widetilde{Q}_{4}+Q)\}

(13)

By condition (11), $Q$ is a well defined positive constant and is clearly independent of $c$ . Therefore $K_{2}$ is also positive and decreases to a positive constant as $c\to 0$ .

2.3 Statement of Results

We now give our two main results on the properties of hyperbolic coordinates for quasi-hyperbolic orbits.

2.3.1 Convergence of hyperbolic coordinates

Our first result concerns the dependence of hyperbolic coordinates on the iterate $k$ . Notice that a-priori there need not be any relation at all between the hyperbolic coordinates at time $k$ and at time $k+1$ . Indeed, recall from Section 2.1 that $e^{(k)},f^{(k)}$ are the pre-images of semi-axes of the ellipse $\mathcal{S}_{k}=D\Phi^{k}(\mathcal{S})$ and $e^{(k+1)},f^{(k+1)}$ are the pre-images semi-axes of the ellipse $\mathcal{S}_{k+1}=D\Phi^{k+1}(\mathcal{S})$ . Since $\mathcal{S}_{k+1}=D\Phi_{\xi_{k}}(\mathcal{S}_{k})$ , it is easy to construct examples in which the major and minor axes of $\mathcal{S}_{k}$ are mapped by $D\Phi_{\xi_{k}}$ to pretty much any desired position in $\mathcal{S}_{k+1}$ . An extreme case would be for $D\Phi_{\xi_{k}}$ to map the major (resp. minor) axis of $\mathcal{S}_{k}$ to the minor (resp. major) axis of $\mathcal{S}_{k+1}$ , implying that the most contracting (resp. most expanding) vector under $D\Phi^{k}$ is the most expanded (resp; most contracted) vector by $D\Phi^{k+1}$ , in which case we have $e^{(k+1)}=f^{(k)}$ and $f^{(k+1)}=e^{(k)}$ .

This shows that in principle hyperbolic coordinates can change wildly for different values of $k$ , which can make it very difficult to use them in any effective way. However, there are (at least) two ways to control such “erratic” changes. The first is by assuming the existence of some “hyperbolic conefield” that guarantees that at every step the derivative maps “expanding directions” to “expanding directions”, thus avoiding the possibility of “switching” the most contracted and most expanded vectors as described above. The existence of such conefields, however, is a quite strong assumption, which is not generally satisfied. A more general approach, and the focus of our results, is based on the observation that if the co-eccentricity of $D\Phi^{k}$ is very small, then the ellipse $\mathcal{S}_{k}$ is very “thin” (in the sense that the ratio between the minor and major axes is very small), and $D\Phi_{\xi_{k}}$ would have to have even smaller co-eccentricity to switch the contracting and expanding directions since it would have to map the minor axis of $\mathcal{S}_{k}$ to a vector whose norm is larger than the image of the major axis. Some of the conditions on the definitions of quasi-hyperbolicity are precisely motivated by the use of this approach in order to control the fluctuation of the hyperbolic coordinates. We will prove the following.

Theorem 2.4.

There are constants $Q_{1},\widetilde{Q}_{1}$ such that for every $k\geq 1$ and every $1\leq i\leq k$ , if $\xi_{0}$ is quasi-hyperbolic up to time $k$ of type (I), then

\|e^{(k)}-e^{(i)}\|\leq Q_{1}\left(\frac{\Gamma\widetilde{\Gamma}c}{\lambda}\right)^{i},

(14)

while if $\xi_{0}$ is quasi-hyperbolic up to time $k$ of type (II), then

\|e^{(k)}-e^{(i)}\|\leq\widetilde{Q}_{1}\left(\frac{c}{\tilde{c}}\right)^{i}.

(15)

In particular, in the non-singular setting, where we can take $\tilde{c}=1$ , we have

\|e^{(k)}-e^{(i)}\|\leq\widetilde{Q}_{1}c^{i}

(16)

Remark 2.15.

Condition (I) says that $c<(\lambda/\Gamma\widetilde{\Gamma})^{3}<1$ which implies $c<\lambda/\Gamma\widetilde{\Gamma}$ and therefore $\Gamma\widetilde{\Gamma}c/\lambda<1$ , and Condition (II) says that $c<\tilde{c}$ , and therefore all 3 in (14), (15), (16), are decreasing exponentially in $i$ .

Remark 2.16.

The expression (16) captures, in its simplest form, the “spirit” of this result and, to some extent, the main motivation for the definition of quasi-hyperbolicity. Since $c\in(0,1)$ , this implies that the sequence of hyperbolic coordinates form a Cauchy sequence and therefore converge as $k\to\infty$ as long as $\xi_{0}$ is quasi-hyperbolic for all $k\geq 1$ . Conditions (14)-(15) imply the same in the singular case.

Remark 2.17.

A bound similar to (16) was proved in [3, 16, 22] in terms of the bound $b$ for the determinant. In these papers the determinant is always assumed to be small and therefore this bound would not apply to certain systems, for example to area-preserving maps. We have here that the determinant is not in fact the natural quantity to bound this convergence but rather the co-eccentricity, which can be $<1$ , and possibly very small, even for area-preserving maps.

Remark 2.18.

The bounds in (14) and (15) are formulated in terms of $i$ . This means that no matter how large $k\geq i$ is, as long as $\xi_{0}$ is quasi-hyperbolic up to time $k$ for the same given set of constants $\mathfrak{C}$ , the hyperbolic coordinates of order $k$ must remain within a fixed “cone” around the hyperbolic coordinates of order $i$ . In particular, if we can compute or estimate the direction $e^{(1)}$ using the explicit formula (5) then (14) and (15) give bounds on the possible positions of all “future” contracting directions $e^{(k)}$ .

Remark 2.19.

The results above are stated for the most contracting directions $e^{(i)},e^{(k)}$ but since hyperbolic coordinates are always orthogonal, exactly the same statements clearly hold for $f^{(i)},f^{(k)}$ .

2.3.2 Derivative of hyperbolic coordinates

To introduce our second main result, recall the expression in (5), which shows that the hyperbolic coordinates depend $C^{1}$ on the base point $\xi_{0}$ . We can therefore consider the derivatives $De^{(k)}$ and $Df^{(k)}$ of the hyperbolic coordinates with respect to the base point (notice that $De^{(k)}=Df^{(k)}$ since $e^{(k)}$ and $f^{(k)}$ are always orthogonal). This derivative, and in particular the norm of this derivative, is of interest as it has several implications, for example for the geometry of the local foliations given by the integral curves of the unit vector fields defined by $e^{(k)}$ and $f^{(k)}$ (recall the discussion in Section 2.1.3). We show that this norm is uniformly bounded in $k$ by a constant that essentially depends on on the constant $c$ which bounds the co-eccentricity.

Theorem 2.5.

There are constants $K_{1}$ and $K_{2}$ such that for every $k\geq 1$ , if $\xi_{0}$ is a quasi-hyperbolic point up to time $k$ , then for $\varsigma=x,y$ , we have

\|D_{\xi_{0}}e^{(k)}\|\leq K_{1}\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|+K_{2}c\leq K_{1}\sqrt{2}\|\partial_{\varsigma}D\Phi_{\xi_{0}}e^{(1)}\|+K_{2}c.

(17)

Furthermore, $K_{1}$ and $K_{2}$ monotonically decrease to nonzero constants as $c\to 0$ .

Remark 2.20.

We emphasize that the constants $K_{1}$ and $K_{2}$ , defined explicitly in (12) and (13) above, are independent of $k$ and just depend on the constants in $\mathfrak{C}$ and on $B,\tilde{B},C,D$ in the definition of quasi-hyperbolicity. In particular, the variation of the hyperbolic coordinates of arbitrarily high order is uniformly bounded.

Remark 2.21.

The first term $\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|$ in (17) involves the second derivative of the map $\Phi$ . It is the operator norm for the linear map $v\mapsto D^{2}\Phi_{\xi_{0}}\left(e^{(1)},v\right)$ . In other words, this describes the variation of the action of $D\Phi_{\xi}$ on the vector field $e^{(1)}(\xi)$ . It depends only on the first iterate of $\Phi$ and is coordinate-free as its formulation does not presuppose any a-priori choice of coordinate systems. In practice, however, estimating $\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|$ may require working in some specific choice of coordinates in which case the second bound in (17) is more useful. Indeed, we can then use information about the first order partial derivatives of $\Phi$ to estimate the position of $e^{(1)}$ using (5) and then information about the second order partial derivatives to estimate $\|\partial_{\varsigma}D\Phi_{\xi_{0}}e^{(1)}\|$ . For example, if we choose a coordinate system where $D\Phi$ is “mostly contracting” in the vertical direction, so that $f^{(1)}\approx(1,0)$ and $e^{(1)}\approx(0,1)$ , then $D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)$ is a linear map approximated by the matrix

D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\approx\left(\begin{array}[]{cc}\partial_{xy}\Phi_{1}(\xi_{0})&\partial_{yy}\Phi_{1}(\xi_{0})\\ \partial_{xy}\Phi_{2}(\xi_{0})&\partial_{yy}\Phi_{2}(\xi_{0})\end{array}\right)=\partial_{y}(D\Phi_{\xi_{0}})

(18)

If the map $\Phi$ is a $C^{2}$ perturbation of a one dimensional map, as in the strongly dissipative Hénon maps of [3], then $\partial_{\varsigma y}\Phi_{j}(\xi_{0})$ is small for $\varsigma=x,y$ and $j=1,2$ and $\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|$ is bounded by a small constant.

Remark 2.22.

The second term $K_{2}c$ in (17) is arguably the most important part of the statement as it highlights the significance of the co-eccentricity constant $c$ . Previous estimates of the variation of hyperbolic coordinates have always been formulated in terms of the bound $b$ for the determinant, and moreover have assumed that this bound was “sufficiently small”. A main innovation in our results is to observe that the co-eccentricity is the key quantity in these estimates, not the determinant. In particular this allows us to apply the results to systems in which the determinant is not necessarily small, even area-preserving systems.

2.4 Overview of the Proof

In Section 3 we prove Theorem 2.4, see Propositions 3.3 and 3.4. In Section 4 we discuss how to bound the term $\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|$ in specific coordinate systems, thus proving the second bound in Theorem 2.5. In Section 5 we give some a-priori bounds for the variation of hyperbolic coordinates, and in Section 6, we take advantage of the quasi-hyperbolicity conditions to turn those abstract a-priori bounds into concrete bounds and so complete the proof of the first bound of Theorem 2.5.

3 Convergence of Hyperbolic Coordinates

In this section we prove Theorem 2.4. In Section 3.1) we prove a-priori bounds on $\|e^{(k)}-e^{(i)}\|$ and $\|e^{(k)}_{i}\|$ that do not assume any hyperbolicity at all apart form the existence of hyperboloic coordinates. We then use these estimates to find more explicit bounds assuming conditions (I) and condition (II) in Definition 2.3.

3.1 A priori bounds

We recall the definition of co-eccentiricity in (2) and let

\widetilde{C}_{\xi_{0},k}=\max_{1\leq i\leq k}\sqrt{\frac{2}{1-C_{\xi_{0},i}^{2}}}.

(19)

We note that the co-eccentricity of a sequence of linear maps, unlike the determinant, is not multiplicative. So there need be no relationship between product or sum of the pointwise single-step co-eccentricities $C_{\xi_{j},1}$ and the accumulated eccentricity $C_{\xi_{0},k}$ . In particular, $C_{\xi_{0},k}$ need not be monotone in $k$ . For the next two lemmas we just suppose that $\xi_{0}$ is a point at which hyperbolic coordinates of order $i$ are defined for all $1\leq i\leq k$ .

Lemma 3.1.

For every $1\leq i\leq k$

$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\widetilde{C}_{\xi_{0},k}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|};$	(20)
$\displaystyle\\|e^{(k)}_{i}\\|$	$\displaystyle\leq\\|(D\Phi_{\xi_{0}}^{i})^{-1}\\|^{-1}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|};$	(21)
$\displaystyle\frac{\\|e^{(k)}_{i}\\|}{\|\det D\Phi_{\xi_{0}}^{i}\|}$	$\displaystyle\leq\frac{1}{\\|D\Phi_{\xi_{0}}^{i}\\|}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{\|\det D\Phi_{\xi_{i}}^{j-i}\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{0}}^{j+1}\\|}.$	(22)

A less sharp but more elegant set of bounds can be obtained by recalling that

C_{\xi_{j},1}:=\frac{\|(D\Phi_{\xi_{j}})^{-1}\|^{-1}}{\|D\Phi_{\xi_{j}}\|}\quad\text{and lettiing}\quad\mathcal{T}_{\xi_{0},i}^{(k)}:=\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}}{C_{\xi_{j},1}}.

(23)

We can then show the following.

Lemma 3.2.

For every $1\leq i\leq k$

$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\mathcal{T}_{\xi_{0},i}^{(k)}\widetilde{C}_{\xi_{0},k};$	(24)
$\displaystyle\\|e^{(k)}_{i}\\|$	$\displaystyle\leq\\|(D\Phi_{\xi_{0}}^{i})^{-1}\\|^{-1}+\\|D\Phi_{\xi_{0}}^{i}\\|\mathcal{T}_{\xi_{0},i}^{(k)}\widetilde{C}_{\xi_{0},k};$	(25)
$\displaystyle\frac{\\|e^{(k)}_{i}\\|}{\|\det D\Phi_{\xi_{0}}^{i}\|}$	$\displaystyle\leq\frac{1}{\\|D\Phi_{\xi_{0}}^{i}\\|}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{\|\det D\Phi_{\xi_{i}}^{j-i}\|}{\\|D\Phi_{\xi_{0}}^{j}\\|^{2}C_{\xi_{j},1}}.$	(26)

Each of these sets of estimates will be used when assuming either Condition (I) or (II) in the Definition 2.3 of quasi-hyperbolicity. Assuming (I) it will be more convenient to apply Lemma 3.1 whereas assuming (II) it will be more convenient to use Lemma 3.2.

Proof of Lemma 3.1.

To estimate $\|e^{(k)}-e^{(i)}\|$ , we write $\|e^{(k)}-e^{(i)}\|\leq\sum_{j=i}^{k-1}\|e^{(j+1)}-e^{(j)}\|$ and estimate $\|e^{(j+1)}-e^{(j)}\|$ for $j\in\{i,...,k-1\}$ . We write

e^{(j)}=\cos\theta e^{(j+1)}+\sin\theta f^{(j+1)}

(27)

for some $\theta=\theta_{j}$ , $|\theta|\leq\pi/2$ , which implies

\|e^{(j+1)}-e^{(j)}\|=\big{(}(1-\cos\theta)^{2}+\sin^{2}\theta\big{)}^{1/2}\leq\sqrt{2}|\sin\theta|.

(28)

By orthogonality of $\{e^{(j+1)}_{j+1},f^{(j+1)}_{j+1}\}$ , after applying $D\Phi_{\xi_{0}}^{j}$ to both sides of (27) and taking the norm, we get

\|e^{(j)}_{j+1}\|^{2}=\cos^{2}\theta\|e^{(j+1)}_{j+1}\|^{2}+\sin^{2}\theta\|f^{(j+1)}_{j+1}\|^{2}.

This implies

\sin^{2}\theta=\frac{\left(\frac{\|e^{(j)}_{j+1}\|}{\|f^{(j+1)}_{j+1}\|}\right)^{2}-\left(\frac{\|e^{(j+1)}_{j+1}\|}{\|f^{(j+1)}_{j+1}\|}\right)^{2}}{1-\left(\frac{\|e^{(j+1)}_{j+1}\|}{\|f^{(j+1)}_{j+1}\|}\right)^{2}}\leq\frac{\left(\frac{\|e^{(j)}_{j+1}\|}{\|f^{(j+1)}_{j+1}\|}\right)^{2}}{1-\left(\frac{\|e^{(j+1)}_{j+1}\|}{\|f^{(j+1)}_{j+1}\|}\right)^{2}}.

Notice that

\|e^{(j)}_{j+1}\|\leq\|D\Phi_{\xi_{j}}\|\|e^{(j)}_{j}\|=\|D\Phi_{\xi_{j}}\|\|(D\Phi_{\xi_{0}}^{j})^{-1}\|^{-1}\quad\textrm{and}\quad\|f^{(j+1)}_{j+1}\|=\|D\Phi_{\xi_{0}}^{j+1}\|,

and also that

\frac{\|e^{(j+1)}_{j+1}\|}{\|f^{(j+1)}_{j+1}\|}=C_{\xi_{0},j+1}.

Using also the fact that $\|(D\Phi_{\xi_{0}}^{j})^{-1}\|^{-1}=C_{\xi_{0},j}\|D\Phi_{\xi_{0}}^{j}\|$ , we obtain

	$\displaystyle\sin^{2}\theta$	$\displaystyle\leq\frac{\left(\frac{\\|e^{(j)}_{j+1}\\|}{\\|f^{(j+1)}_{j+1}\\|}\right)^{2}}{1-\left(\frac{\\|e^{(j+1)}_{j+1}\\|}{\\|f^{(j+1)}_{j+1}\\|}\right)^{2}}\leq\frac{1}{1-C^{2}_{\xi_{0},j+1}}\frac{\\|D\Phi_{\xi_{j}}\\|^{2}\\|(D\Phi_{\xi_{0}}^{j})^{-1}\\|^{-2}}{\\|D\Phi_{\xi_{0}}^{j+1}\\|^{2}}$		(29)
		$\displaystyle=\frac{1}{1-C^{2}_{\xi_{0},j+1}}\frac{C_{\xi_{0},j}^{2}\\|D\Phi_{\xi_{0}}^{j}\\|^{2}\\|D\Phi_{\xi_{j}}\\|^{2}}{\\|D\Phi_{\xi_{0}}^{j+1}\\|^{2}}$		(29)

Therefore, (28) and (29) give us:

	$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\sum_{j=i}^{k-1}\\|e^{(j)}-e^{(j+1)}\\|$
		$\displaystyle\leq\sqrt{\frac{2}{1-C_{\xi_{0},i}^{2}}}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|}$
		$\displaystyle\leq\widetilde{C}_{\xi_{0},k}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|}.$

This gives us (20). To prove (21), we use (20) to show:

\|e^{(k)}_{i}\|\leq\|e^{(i)}_{i}\|+\|D\Phi_{\xi_{0}}^{i}\|\|e^{(k)}-e^{(i)}\|\leq\|(D\Phi_{\xi_{0}}^{i})^{-1}\|^{-1}+\widetilde{C}_{\xi_{0},k}\|D\Phi_{\xi_{0}}^{i}\|\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j+1}\|}.

Finally, noting $C_{\xi_{0},j}\|D\Phi_{\xi_{0}}^{j}\|/|\det D\Phi_{\xi_{0}}^{j}|=\|D\Phi_{\xi_{0}}^{j}\|^{-1}$ , factoring out $|\det D\Phi_{\xi_{0}}^{i}|$ from the summands in (21) gives us:

\frac{C_{\xi_{0},j}\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j+1}\|}=\frac{|\det D\Phi_{\xi_{0}}^{i}||\det D\Phi_{\xi_{i}}^{j-i}|}{|\det D\Phi_{\xi_{0}}^{j}|}\frac{C_{\xi_{0},j}\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j+1}\|}=|\det D\Phi_{\xi_{0}}^{i}|\frac{|\det D\Phi_{\xi_{i}}^{j-i}|\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{0}}^{j+1}\|},

and since $\|(D\Phi_{\xi_{0}}^{i})^{-1}\|^{-1}/|\det D\Phi_{\xi_{0}}^{i}|=\|D\Phi_{\xi_{0}}\|^{-1}$ , dividing (21) by $|\det D\Phi_{\xi_{0}}^{i}|$ gives us (22). ∎

Proof of Lemma 3.2.

Observe first of all that we always have $\|D\Phi_{\xi_{0}}^{j+1}\|\geq\|D\Phi_{\xi_{0}}^{j}\|\|(D\Phi_{\xi_{j}})^{-1}\|^{-1}$ and so

\frac{\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j+1}\|}\leq\frac{1}{C_{\xi_{j},1}}\quad\text{ and }\quad\frac{\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{0}}^{j+1}\|}\leq\frac{1}{\|D\Phi_{\xi_{0}}^{j}\|^{2}C_{\xi_{j},1}}

(30)

Substituting the first inequality into (20) and (21) gives (24) and (25) respectively, and substituting the second inequality into (22) gives (26). ∎

Remark 3.1.

The first step in the proof of Lemma 3.1 is to use the triangle inequality to write $\|e^{(k)}-e^{(i)}\|\leq\sum_{j=i}^{k-1}\|e^{(j+1)}-e^{(j)}\|$ , and then to estimate each term $\|e^{(j+1)}-e^{(j)}\|$ . Strictly speaking, this first step is not necessary to obtain an a priori bound; one could use the same arguments to directly estimate $\|e^{(k)}-e^{(i)}\|$ instead. Doing so would, for example, give us the bound

\|e^{(k)}-e^{(i)}\|\leq\widetilde{C}_{\xi_{0},k}\frac{C_{\xi_{0},i}\|D\Phi_{\xi_{0}}^{i}\|\|D\Phi_{\xi_{i}}^{k-i}\|}{\|D\Phi_{\xi_{i}}^{k}\|}

(31)

instead of the bound in (20), and similar alternative bounds to (21) and (22) can also be derived. Rather than in terms of the sum

\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\|D\Phi_{\xi_{0}}^{j}\|\|D\Phi_{\xi_{j}}\|}{\|D\Phi_{\xi_{0}}^{j+1}\|},

(32)

we instead get a bound in terms of the quotient

\frac{C_{\xi_{0},i}\|D\Phi_{\xi_{0}}^{i}\|\|D\Phi_{\xi_{i}}^{k-i}\|}{\|D\Phi_{\xi_{0}}^{k}\|}.

(33)

The former takes the sum of the estimates for each transition from $\xi_{j}$ to $\xi_{j+1}$ under $\Phi$ , whereas the latter estimates directly the transition from $\xi_{i}$ to $\xi_{k}$ under $\Phi^{k-i}$ . The main reason we estimate the sum in (32) instead of the term in (33) becomes apparent only after introducing quasi-hyperbolicity assumptions. We will see in Propositions 3.3 and 3.4 that $\|e^{(k)}-e^{(i)}\|$ and $\|e^{(k)}_{i}\|$ both have upper bounds that are exponential in $i$ but independent of $k$ . This is because we will approximate the sum in (32) with the tail of a geometric series, which decays exponentially with $i$ and is independent of $k$ . However, the expression in (33) cannot be given an upper bound independent of $k$ without introducing much stronger restrictions than quasi-hyperbolicity.

3.2 Convergence with hyperbolicity assumptions

We now estimate $\|e^{(k)}-e^{(i)}\|$ and $\|e^{(k)}_{i}\|$ applying the bounds given in the Definition 2.3 of quasi-hyperbolicity, treating separately the situations in which Condition (I) and Condition (II) are satisfied (sections 3.2.1 and 3.2.2 respectively). Note that the a priori estimates in Lemma 3.1 are stronger than the estimates in Lemma 3.2. However, the conclusions we obtain from these lemmas, which are formulated in Propositions 3.3 and 3.4 respectively, are not similarly related: it is not immediate that one set of estimates is stronger than the other. This is because we bound different terms in different ways in the two cases.

3.2.1 Convergence with quasi-hyperbolicity of type (I)

First we suppose that the constants satisfy Condition (I) of Definition 2.3 .

Proposition 3.3.

Suppose $\xi_{0}$ is singular quasi-hyperbolic up to time $k$ and satisfies condition (I). Then:

\|e^{(k)}-e^{(i)}\|\leq Q_{1}\left(\frac{\Gamma\widetilde{\Gamma}c}{\lambda}\right)^{i};

(34)

\|e^{(k)}_{i}\|\leq Q_{1}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda}\right)^{i};

(35)

\frac{\|e^{(k)}_{i}\|}{|\det D\Phi_{\xi_{0}}^{i}|}\leq Q_{2}\left(\frac{\Gamma\widetilde{\Gamma}}{\lambda^{2}}\right)^{i}

(36)

Proof.

We first note that by the second set of inequalities in (9) and the fact that $c<1$ ,

\widetilde{C}_{\xi_{0},k}=\max_{1\leq i\leq k}\sqrt{\frac{2}{1-C_{\xi_{0},i}^{2}}}\leq\max_{1\leq i\leq k}\sqrt{\frac{2}{1-B^{2}c^{2i}}}=\sqrt{\frac{2}{1-B^{2}c^{2}}}=Q_{0}.

(37)

So (9), (10), (20), and (37) give us:

$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\widetilde{C}_{\xi_{0},k}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|}$	(38)
	$\displaystyle\leq Q_{0}\sum_{j=i}^{k-1}\frac{BD^{2}c^{j}\Gamma^{j+1}\widetilde{\Gamma}^{j}}{C\lambda^{j+1}}$
	$\displaystyle\leq\frac{Q_{0}BD^{2}\Gamma}{C\lambda}\sum_{j=i}^{\infty}\left(\frac{\Gamma\widetilde{\Gamma}c}{\lambda}\right)^{j}$
	$\displaystyle=\frac{Q_{0}BD^{2}\Gamma}{C\lambda}\frac{1}{1-\frac{\Gamma\widetilde{\Gamma}c}{\lambda}}\left(\frac{\Gamma\widetilde{\Gamma}c}{\lambda}\right)^{i}$
	$\displaystyle=\frac{Q_{0}BD^{2}\Gamma}{C(\lambda-\Gamma\widetilde{\Gamma}c)}\left(\frac{\Gamma\widetilde{\Gamma}c}{\lambda}\right)^{i}.$

The first equality follows because $\Gamma\widetilde{\Gamma}c<\lambda$ as a consequence of (I). So (34) now follows. Next, note:

\|(D\Phi_{\xi_{0}}^{i})^{-1}\|^{-1}=\|D\Phi_{\xi_{0}}^{i}\|C_{\xi_{0},i}\leq BD(\Gamma c)^{i}.

(39)

So from (21), (39), and (38), we obtain:

	$\displaystyle\\|e^{(k)}_{i}\\|$	$\displaystyle\leq\\|(D\Phi_{\xi_{0}}^{i})^{-1}\\|^{-1}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|}$
		$\displaystyle\leq BD(\Gamma c)^{i}+\frac{Q_{0}BD^{3}\Gamma}{C(\lambda-\Gamma\widetilde{\Gamma}c)}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda}\right)^{i}$
		$\displaystyle\leq\left(BD+\frac{Q_{0}BD^{3}\Gamma}{C(\lambda-\Gamma\widetilde{\Gamma}c)}\right)\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda}\right)^{i}$
		$\displaystyle=Q_{1}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda}\right)^{i}.$

Note the final inequality follows because $\Gamma c\leq\Gamma^{2}\widetilde{\Gamma}c/\lambda$ because $\Gamma\geq\lambda$ and $\widetilde{\Gamma}\geq 1$ by (I). Finally, applying (9), (10), and (37) to (22), we obtain:

	$\displaystyle\frac{\\|e^{(k)}_{i}\\|}{\|\det D\Phi_{\xi_{0}}^{i}\|}$	$\displaystyle\leq\frac{1}{\\|D\Phi_{\xi_{0}}^{i}\\|}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{\|\det D\Phi_{\xi_{i}}^{j-i}\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{0}}^{j+1}\\|}$
		$\displaystyle\leq\frac{1}{C\lambda^{i}}+Q_{0}D\Gamma^{i}\sum_{j=i}^{k-1}\frac{b^{j-i}D\Gamma\widetilde{\Gamma}^{j}}{C^{2}\lambda^{2j+1}}$
		$\displaystyle\leq\frac{1}{C\lambda^{i}}+\frac{Q_{0}D^{2}\Gamma}{C^{2}\lambda}\frac{\Gamma^{i}}{b^{i}}\sum_{j=i}^{\infty}\left(\frac{\widetilde{\Gamma}b}{\lambda^{2}}\right)^{j}$
		$\displaystyle=\frac{1}{C\lambda^{i}}+\frac{Q_{0}D^{2}\Gamma}{C^{2}\lambda}\frac{1}{1-\frac{\widetilde{\Gamma}b}{\lambda^{2}}}\frac{\Gamma^{i}}{b^{i}}\left(\frac{\widetilde{\Gamma}b}{\lambda^{2}}\right)^{i}$
		$\displaystyle=\frac{1}{C\lambda^{i}}+\frac{Q_{0}D^{2}\Gamma\lambda}{C^{2}(\lambda^{2}-\widetilde{\Gamma}b)}\left(\frac{\Gamma\widetilde{\Gamma}}{\lambda^{2}}\right)^{i}$
		$\displaystyle\leq\left(\frac{1}{C}+\frac{Q_{0}D^{2}\Gamma\lambda}{C^{2}(\lambda^{2}-\widetilde{\Gamma}b)}\right)\left(\frac{\Gamma\widetilde{\Gamma}}{\lambda^{2}}\right)^{i}$
		$\displaystyle=Q_{2}\left(\frac{\Gamma\widetilde{\Gamma}}{\lambda^{2}}\right)^{i}.$

The final inequality holds because $\Gamma\widetilde{\Gamma}/\lambda^{2}\geq 1/\lambda$ , since $\Gamma\geq\lambda$ and $\widetilde{\Gamma}\geq 1$ . ∎

3.2.2 Convergence with quasi-hyperbolicity of type (II)

We now suppose the constants satisfy condition (II) in Definition 2.3.

Proposition 3.4.

Suppose $\xi_{0}$ is singular hyperbolic up to time $k$ and satisfies (II). Then:

\|e^{(k)}-e^{(i)}\|\leq\widetilde{Q}_{1}\left(\frac{c}{\tilde{c}}\right)^{i};

(40)

\|e^{(k)}_{i}\|\leq\widetilde{Q}_{1}\left(\frac{\Gamma c}{\tilde{c}}\right)^{i};

(41)

\frac{\|e^{(k)}_{i}\|}{|\det D\Phi_{\xi_{0}}^{i}|}\leq\widetilde{Q}_{2}\left(\frac{\Gamma}{\lambda^{2}\tilde{c}}\right)^{i}.

(42)

Proof of Proposition 3.4.

The proof essentially consists of applying the estimates in Definition 2.3 and in (II) to the a priori estimates in Lemma 3.2. Observe first that since $c<\tilde{c}$ ,

\mathcal{T}^{(k)}_{\xi_{0},i}=\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}}{C_{\xi_{j},1}}<\sum_{j=i}^{\infty}\frac{Bc^{j}}{\widetilde{B}\tilde{c}^{j}}=\frac{B}{\widetilde{B}}\frac{1}{1-\frac{c}{\tilde{c}}}\left(\frac{c}{\tilde{c}}\right)^{i}=\frac{B\tilde{c}}{\widetilde{B}(\tilde{c}-c)}\left(\frac{c}{\tilde{c}}\right)^{i}.

(43)

Note (37) also applies in this setting. By (37) and (43), we get:

\mathcal{T}^{(k)}_{\xi_{0},i}\widetilde{C}_{\xi_{0},k}\leq\frac{Q_{0}B\tilde{c}}{\widetilde{B}(\tilde{c}-c)}\left(\frac{c}{\tilde{c}}\right)^{i}.

(44)

Now (40) follows from (24) and (44). Meanwhile, by (25) and (44), using again that $\|(D\Phi_{\xi_{0}}^{i})^{-1}\|^{-1}\leq BD(\Gamma c)^{i}$ by (39), we get:

	$\displaystyle\\|e^{(k)}_{i}\\|$	$\displaystyle\leq\\|(D\Phi_{\xi_{0}}^{i})^{-1}\\|^{-1}+\\|D\Phi_{\xi_{0}}^{i}\\|\mathcal{T}_{\xi_{0},i}^{(k)}\widetilde{C}_{\xi_{0},k}$
		$\displaystyle\leq BD(\Gamma c)^{i}+\frac{Q_{0}B\tilde{c}}{\widetilde{B}(\tilde{c}-c)}\left(\frac{\Gamma c}{\tilde{c}}\right)^{i}$
		$\displaystyle\leq\left(BD+\frac{Q_{0}B\tilde{c}}{\widetilde{B}(\tilde{c}-c)}\right)\left(\frac{\Gamma c}{\tilde{c}}\right)^{i}$
		$\displaystyle=\widetilde{Q}_{1}\left(\frac{\Gamma c}{\tilde{c}}\right)^{i}.$

The final inequality holds because $\tilde{c}<1$ . This proves (41). Finally, if (II) holds, then by applying (9) and (10) to (26), we obtain:

	$\displaystyle\frac{\\|e^{(k)}_{i}\\|}{\|\det D\Phi_{\xi_{0}}^{i}\|}$	$\displaystyle\leq\frac{1}{\\|D\Phi_{\xi_{0}}^{i}\\|}+\\|D\Phi_{\xi_{0}}^{i}\\|\widetilde{C}_{\xi_{0},k}\sum_{j=i}^{k-1}\frac{\|\det D\Phi_{\xi_{i}}^{j-i}\|}{\\|D\Phi_{\xi_{0}}^{j}\\|^{2}C_{\xi_{j},1}}$
		$\displaystyle\leq\frac{1}{C\lambda^{i}}+Q_{0}D\Gamma^{i}\sum_{j=i}^{k-1}\frac{b^{j-i}}{C^{2}\lambda^{2j}\widetilde{B}\tilde{c}^{j}}$
		$\displaystyle\leq\frac{1}{C\lambda^{i}}+\frac{Q_{0}D}{C^{2}\widetilde{B}}\frac{\Gamma^{i}}{b^{i}}\sum_{j=i}^{\infty}\left(\frac{b}{\lambda^{2}\tilde{c}}\right)^{j}$
		$\displaystyle=\frac{1}{C\lambda^{i}}+\frac{Q_{0}D}{C^{2}\widetilde{B}}\frac{1}{1-\frac{b}{\lambda^{2}\tilde{c}}}\frac{\Gamma^{i}}{b^{i}}\left(\frac{b}{\lambda^{2}\tilde{c}}\right)^{i}$
		$\displaystyle=\frac{1}{C\lambda^{i}}+\frac{Q_{0}\lambda^{2}\tilde{c}}{C^{2}\widetilde{B}(\lambda^{2}\tilde{c}-b)}\left(\frac{\Gamma}{\lambda^{2}\tilde{c}}\right)^{i}$
		$\displaystyle\leq\left(\frac{1}{C}+\frac{Q_{0}D\lambda^{2}\tilde{c}}{C^{2}\widetilde{B}(\lambda^{2}\tilde{c}-b)}\right)\left(\frac{\Gamma}{\lambda^{2}\tilde{c}}\right)^{i}$
		$\displaystyle=\widetilde{Q}_{2}\left(\frac{\Gamma}{\lambda^{2}\tilde{c}}\right)^{i}.$

The first equality holds by the second inequality in (II), and the final inequality holds since $\Gamma/\lambda^{2}\tilde{c}\geq 1/\lambda$ , because $\tilde{c}\leq 1$ and because $\Gamma\geq\lambda$ by (11). ∎

Proof of Theorem 2.4.

Theorem 2.4 follows immediately from (34) and (40). ∎

4 Slow Variation in Local Coordinates

We are now ready to start the proof of Theorem 2.5. In this section we prove the second inequality in (17), which gives a more explicit bound for a given choice of local coordinates. This is relatively simple and quite general and contains a couple of bounds which we will use again in the following sections.

Proposition 4.1.

In normal coordinates based at $\xi_{0}$ , the norm $\|D^{2}\Phi_{\xi_{0}}\|$ satisfies:

\max_{\varsigma=x,y}\{\|\partial_{\varsigma}(D\Phi_{\xi_{0}})\|\}\leq\|D^{2}\Phi_{\xi_{0}}\|\leq\sqrt{2}\max_{\varsigma=x,y}\{\|\partial_{\varsigma}(D\Phi_{\xi_{0}})\|\}.

(45)

Moreover, for any tangent vector $v$ at $\xi_{0}$ :

\max_{\varsigma=x,y}\left\{\left\|\left[\partial_{\varsigma}\left(D\Phi_{\xi_{0}}\right)\right]v\right\|\right\}\leq\left\|D^{2}\Phi_{\xi_{0}}\left(v,\cdot\right)\right\|\leq\sqrt{2}\max_{\varsigma=x,y}\left\{\left\|\left[\partial_{\varsigma}\left(D\Phi_{\xi_{0}}\right)\right]v\right\|\right\}.

(46)

Indeed, letting $v=e^{(1)}$ and substituting the second inequality of (46) into (17) we get the required estimate.

We will prove a more general version of Proposition 4.1 in higher dimensions. Specifically, let $\Phi:M\to M$ be a $C^{2}$ map of a Riemannian $n$ -manifold $M$ , and let $(x^{1},\ldots,x^{n})$ be Riemannian normal coordiantes at $\xi_{0}$ . For each $k=1,\ldots,n$ , we define the second-order partial derivative with respect to $x^{k}$ to be the linear map $\partial_{x^{k}}(D\Phi_{\xi_{0}}):T_{\xi_{0}}M\to T_{\Phi(\xi_{0})}M$ given by:

\partial_{x^{k}}(D\Phi_{\xi_{0}})=\begin{pmatrix}\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{1}\partial x^{k}}(\xi_{0})&\cdots&\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{n}\partial x^{k}}(\xi_{0})\\ \vdots&&\vdots\\ \displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{1}\partial x^{k}}(\xi_{0})&\cdots&\displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{n}\partial x^{k}}(\xi_{0})\end{pmatrix}.

(47)

Proposition 4.2.

For all $v\in T_{\xi_{0}}M$ , in normal coordiantes at $\xi_{0}$ ,

\max_{1\leq k\leq n}\|[\partial_{x^{k}}(D\Phi_{\xi_{0}})]v\|\leq\|D^{2}\Phi_{\xi_{0}}(v,\cdot)\|\leq\sqrt{n}\max_{1\leq k\leq n}\|[\partial_{x^{k}}(D\Phi_{\xi_{0}})]v\|.

(48)

In particular,

\max_{1\leq k\leq n}\|\partial_{x^{k}}(D\Phi_{\xi_{0}})\|\leq\|D^{2}\Phi_{\xi_{0}}\|\leq\sqrt{n}\max_{1\leq k\leq n}\|\partial_{x^{k}}(D\Phi_{\xi_{0}})\|.

(49)

This clearly implies Proposition 4.1 when $n=2$ .

To prove Proposition 4.2 we will prove a series of lemmas, most of which are linear-algebraic observations. Notice the middle term $D^{2}\Phi_{\xi_{0}}$ is in fact a bilinear map $T_{\xi_{0}}M\times T_{\xi_{0}}M\to T_{\Phi(\xi_{0})}M$ . We begin by introducing notation to study the coordinates of $D^{2}\Phi_{\xi_{0}}$ and show that the operator $\partial_{x^{k}}(D\Phi_{\xi_{0}})$ in (47) in fact the monolinear map $D^{2}\Phi_{\xi_{0}}(\cdot,\partial_{x^{k}})$ .

Suppose $b:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ is a bilinear form. Given a basis $\mathcal{B}=\{\mathbf{u}_{1},\ldots,\mathbf{u}_{n}\}$ of $\mathbb{R}^{n}$ , let $\mathcal{B}^{*}=\{\omega^{1},\ldots,\omega^{n}\}$ be the corresponding cobasis of linear functionals on $\mathbb{R}^{n}$ , defined by $\omega^{i}(\mathbf{u}_{j})=\delta_{ij}$ for all $1\leq i,j\leq n$ . Then the bilinear form $b$ can be written as

b=\sum_{1\leq i,j\leq k}b_{ij}\omega^{i}\otimes\omega^{j},

for some real coefficients $b_{ij}\in\mathbb{R}$ . Consider now a bilinear map $B:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ . In any basis $\mathcal{B}=(\mathbf{u}_{1},\ldots,\mathbf{u}_{n})$ of $\mathbb{R}^{n}$ , writing $B=(B^{1},\ldots,B^{n})$ , each $B^{k}$ is a bilinear form. Thus, again letting $\mathcal{B}^{*}=(\omega^{1},\ldots,\omega^{n})$ be the cobasis of $\mathcal{B}$ , we can express $B$ as:

B=\begin{pmatrix}B^{1}\\ \vdots\\ B^{n}\end{pmatrix}=\begin{pmatrix}\sum_{i,j}B_{ij}^{1}\omega^{i}\otimes\omega^{j}\\ \vdots\\ \sum_{i,j}B_{ij}^{n}\omega^{i}\otimes\omega^{j}\end{pmatrix}.

(50)

For a particular $\mathbf{v}\in\mathbb{R}^{n}$ , the map $B(\mathbf{v},\cdot)$ given by $\mathbf{x}\mapsto B(\mathbf{v},\mathbf{x})$ is a linear transformation, and thus has a matrix representation in terms of any basis. Referring to (50) and writing a vector $\mathbf{v}$ in coordinates as $\mathbf{v}=(v^{1},\ldots,v^{n})$ , one sees that:

B(\mathbf{v},\cdot)=\begin{pmatrix}B^{1}(\mathbf{v},\cdot)\\ \vdots\\ B^{n}(\mathbf{v},\cdot)\end{pmatrix}=\begin{pmatrix}\sum_{i,j}B^{1}_{ij}v^{i}\omega^{j}\\ \vdots\\ \sum_{i,j}B^{n}_{ij}v^{i}\omega j\end{pmatrix},

and after expanding this “covector form” of $B(\mathbf{v},\cdot)$ , one sees that the matrix form of $B(\mathbf{v},\cdot)$ is:

B(\mathbf{v},\cdot)=\begin{pmatrix}\sum_{i=1}^{n}B^{1}_{i,1}v^{i}&\cdots&\sum_{i=1}^{n}B^{1}_{i,n}v^{i}\\ \vdots&&\vdots\\ \sum_{i=1}^{n}B^{n}_{i,1}v^{i}&\cdots&\sum_{i=1}^{n}B^{n}_{i,n}v^{i}\end{pmatrix}.

(51)

We will use this covector form to study $D^{2}\Phi$ .

Lemma 4.3.

Let $(x^{1},\ldots,x^{n})$ be coordinates at $\xi_{0}$ , and let $V=V^{1}\partial_{x^{1}}+\cdots+V^{n}\partial_{x^{n}}$ , $V^{i}\in\mathbb{R}$ , be a tangent vector at $\xi_{0}$ . Then:

D^{2}\Phi(V,\cdot)=\begin{pmatrix}\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{1}}V^{i}&\cdots&\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{n}}V^{i}\\ \vdots&&\vdots\\ \displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{1}}V^{i}&\cdots&\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{n}}V^{i}\end{pmatrix}.

(52)

In particular, for any of the coordinate vectors $\partial_{x^{k}}$ :

D^{2}\Phi(V,\partial_{x^{k}})=\begin{pmatrix}\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{k}}V^{i}\\ \vdots\\ \displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{k}}V^{i}\end{pmatrix}=\begin{pmatrix}\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{1}\partial x^{k}}&\cdots&\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{n}\partial x^{k}}\\ \vdots&&\vdots\\ \displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{1}\partial x^{k}}&\cdots&\displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{n}\partial x^{k}}\end{pmatrix}\begin{pmatrix}V^{1}\\ \vdots\\ V^{n}\end{pmatrix}=(\partial_{x^{k}}D\Phi)V.

(53)

Proof.

Given a coordinate system $(x^{1},\ldots,x^{n})$ of $M$ , consider the coordinate tangent frame $(\partial_{x^{1}},\ldots,\partial_{x^{n}})$ and coordinate coframe $(dx^{1},\ldots,dx^{n})$ of differential forms, and write $\Phi$ in coordinates as $\Phi=(\Phi_{1},\ldots,\Phi_{n})$ . At each $\xi\in M$ , the second derivative of $\Phi$ is a bilinear map $(T_{\xi}M)\times(T_{\xi}M)\to T_{\Phi(\xi)}M$ . Expressing $D^{2}\Phi$ in terms of this coordinate system as in (50), we have:

D^{2}\Phi=\begin{pmatrix}\displaystyle\sum_{i,j}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{j}}dx^{i}\otimes dx^{j}\\ \vdots\\ \displaystyle\sum_{i,j}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{j}}dx^{i}\otimes dx^{j}\end{pmatrix}.

(54)

Consider now a vector field $V$ on $M$ (for example $V$ could be the time- $1$ stable direction $e^{(1)}$ for a set of time- $1$ hyperbolic coordinates $\{e^{(1)},f^{(1)}\}$ , as in Definition 2.1). In a unit coordinate system, write $V=V^{1}\partial_{x^{1}}+\cdots+V^{n}\partial_{x^{n}}$ , $V^{k}:M\to\mathbb{R}$ smooth functions. Putting $V$ into one of the arguments in (54) and expressing $D^{2}\Phi(V,\cdot)$ as in (51), we note:

D^{2}\Phi(V,\cdot)=\begin{pmatrix}\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{1}}V^{i}&\cdots&\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{n}}V^{i}\\ \vdots&&\vdots\\ \displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{1}}V^{i}&\cdots&\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{n}}V^{i}\end{pmatrix}.

(55)

Putting a coordinate tangent vector $\partial_{x^{k}}$ into $D^{2}\Phi(V,\cdot)$ gives us:

D^{2}\Phi(V,\partial_{x^{k}})=\begin{pmatrix}\displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{1}}{\partial x^{i}\partial x^{k}}V^{i}\\ \vdots\\ \displaystyle\sum_{i=1}^{n}\frac{\partial^{2}\Phi_{n}}{\partial x^{i}\partial x^{k}}V^{i}\end{pmatrix}=\begin{pmatrix}\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{1}\partial x^{k}}&\cdots&\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{n}\partial x^{k}}\\ \vdots&&\vdots\\ \displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{1}\partial x^{k}}&\cdots&\displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{n}\partial x^{k}}\end{pmatrix}\begin{pmatrix}V^{1}\\ \vdots\\ V^{n}\end{pmatrix}=(\partial_{x^{k}}D\Phi)V,

(56)

where $\partial_{x^{k}}D\Phi=\partial_{x^{k}}(D\Phi_{\xi}):T_{\xi}M\to T_{\Phi(\xi)}M$ (at each $\xi\in M$ where these coordinates are defined) is the linear transformation given by

\partial_{x^{k}}(D\Phi_{\xi})=\begin{pmatrix}\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{1}\partial x^{k}}(\xi)&\cdots&\displaystyle\frac{\partial^{2}\Phi_{1}}{\partial x^{n}\partial x^{k}}(\xi)\\ \vdots&&\vdots\\ \displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{1}\partial x^{k}}(\xi)&\cdots&\displaystyle\frac{\partial^{2}\Phi_{n}}{\partial x^{n}\partial x^{k}}(\xi)\end{pmatrix}.

(57)

∎

In order to bound $\|D^{2}\Phi_{\xi_{0}}\|$ in terms of the derivatives $\partial_{x^{k}}(D\Phi_{\xi_{0}})$ , we will compare $D^{2}\Phi_{\xi_{0}}$ to the monolinear map $D\Phi_{\xi_{0}}(\cdot,\partial_{x^{k}})$ (which is equal to $\partial_{x^{k}}(D\Phi_{\xi_{0}})$ by (53)). We first show how to estimate the norm of a general bilinear map $B:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ in terms of the norms of the maps $B(\mathbf{v},\cdot)$ .

Lemma 4.4.

If $B:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a bilinear map, and $\mathcal{B}=(\mathbf{u}_{1},\ldots,\mathbf{u}_{n})$ is an orthonormal basis of $\mathbb{R}^{n}$ , then for any $\mathbf{v}\in\mathbb{R}^{n}$ :

\max_{1\leq k\leq n}\|B(\mathbf{v},\mathbf{u}_{k})\|\leq\|B(\mathbf{v},\cdot)\|\leq\sqrt{n}\max_{1\leq k\leq n}\|B(\mathbf{v},\mathbf{u}_{k})\|

(58)

Moreover:

\max_{1\leq k\leq n}\|B(\cdot,\mathbf{u}_{k})\|\leq\|B\|\leq\sqrt{n}\max_{1\leq k\leq n}\|B(\cdot,\mathbf{u}_{k})\|

(59)

To prove Lemma 4.4 we first prove a simple statement about linear maps.

Sublemma 4.5.

Let $A$ be an $m\times n$ matrix with real or complex entries, whose columns are $\mathbf{a}_{1},\ldots,\mathbf{a}_{n}$ in terms of an orthonormal basis $\mathcal{B}$ . Then we have:

\max_{1\leq k\leq n}\|\mathbf{a}_{k}\|\leq\|A\|\leq\sqrt{n}\max_{1\leq k\leq n}\|\mathbf{a}_{k}\|.

Proof.

Let $\mathcal{B}=\{\mathbf{u}_{1},\ldots,\mathbf{u}_{n}\}$ be an orthonormal basis of $\mathbb{R}^{n}$ . With respect to this basis, write $A=\begin{pmatrix}\mathbf{a}_{1}&\cdots&\mathbf{a}_{n}\end{pmatrix}$ , with $\mathbf{a}_{k}$ the $k$ th column of $A$ . Then $A\mathbf{u}_{k}=\mathbf{a}_{k}$ for all $k$ . In particular, $\|A\|\geq\|\mathbf{a}_{k}\|$ for all $k$ , giving us the left hand side of the statement in the lemma. Meanwhile, writing $\mathbf{v}=v_{1}\mathbf{u}_{1}+\cdots+v_{n}\mathbf{u}_{n}$ , we have:

\|A\|=\sup_{\|\mathbf{v}\|=1}\|A\mathbf{v}\|\leq\sup_{\|\mathbf{v}\|=1}\sum_{k=1}^{n}\|\mathbf{a}_{k}v_{k}\|\leq\left(\max_{1\leq k\leq n}\|\mathbf{a}_{k}\|\right)\sup_{\|\mathbf{v}\|=1}\left(|v_{1}|+\cdots+|v_{n}|\right).

One can use Lagrange multipliers to show that the function $f(v_{1},\ldots,v_{n})=v_{1}+\cdots+v_{n}$ restricted to $v_{1}^{2}+\cdots+v_{n}^{2}=1$ , $v_{k}\geq 1$ , has maximal value $\sqrt{n}$ . This gives us the second inequality of the lemma. ∎

Proof of Lemma 4.4.

Applying Sublemma 4.5 to the linear map $B(\mathbf{v},\cdot)$ , we obtain (58) immediately. Next, noting $\|B\|=\sup_{\|\mathbf{x}\|=\|\mathbf{y}\|=1}\|B(\mathbf{x},\mathbf{y})\|$ for a bilinear map $B:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ , we clearly have the first inequality in (59). For the second inequality of (59), we use (58) to conclude:

\|B\|=\max_{\|\mathbf{v}\|=1}\|B(\mathbf{v},\cdot)\|\leq\sqrt{n}\max_{\|\mathbf{v}\|=1}\max_{1\leq k\leq n}\|B(\mathbf{v},\mathbf{u}_{k})\|\leq\sqrt{n}\max_{1\leq k\leq n}\|B(\cdot,\mathbf{u}_{k})\|.

∎

Proof of Proposition 4.2.

Both (48) and (49) follow from (58) and (59) applied to $D^{2}\Phi_{\xi_{0}}(v,\cdot)$ and $D^{2}\Phi_{\xi_{0}}$ , respectively, noting $(\partial_{x^{1}}(\xi_{0}),\ldots,\partial_{x^{n}}(\xi_{0}))$ forms an orthonormal basis of $T_{\xi_{0}}M$ and that $D^{2}\Phi_{\xi_{0}}(v,\partial_{x^{k}})=(\partial_{x^{k}}D\Phi_{\xi_{0}})v$ by (53). ∎

5 Slow Variation: A Priori Bounds

We now begin the proof of the first and main bound in the statement of Theorem 2.5. In this section we establish a priori bounds on $\|Df^{(k)}\|$ which just rely on the existence of hyperbolic coordinates. In Section 6 we apply the properties of quasi-hyperbolic points to the a priori estimates to obtain the first inequality in the statement of Theorem 2.5.

We emphasize that the first inequality in Theorem 2.5 is independent of the choice of coordinates. However, for the proof it is natural to use an appropriate coordinate system. Thus for the remainder of this section we fix once and for all a Riemannian coordinate system $(x,y)$ based at $\xi_{0}$ , at which the coordinate tangent vectors $\{\partial_{x}(\xi_{0}),\partial_{y}(\xi_{0})\}$ form an orthonormal basis at $T_{\xi_{0}}M$ , though we emphasize that the constants $K_{1}$ and $K_{2}$ in Theorem 2.5 will not depend on this choice of coordinates. Then, for $\varsigma=x,y$ , we have

\partial_{\varsigma}\left(D\Phi_{\xi_{0}}\right)=\left(\begin{array}[]{cc}\partial_{\varsigma x}\Phi_{1}(\xi_{0})&\partial_{\varsigma y}\Phi_{1}(\xi_{0})\\ \partial_{\varsigma x}\Phi_{2}(\xi_{0})&\partial_{\varsigma y}\Phi_{2}(\xi_{0})\end{array}\right).

(60)

The entries of the matrix (60) are just the second order partial derivatives of $\Phi$ with respect to this coordinate system. For simplicity, we use the following notation:

A_{k}:=\frac{\sqrt{2}\|f^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}\quad\text{ and }\quad B_{k}:=\frac{\sqrt{2}\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}.

\mathfrak{E}^{(k)}_{i}:=\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e^{(k)}_{i}\|\|e^{(k)}_{i+1}\|}{|\det(D\Phi^{i+1}_{\xi_{0}})|}\quad\text{ and }\quad\mathfrak{F}^{(k)}_{i}:=\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\|\|f^{(k)}_{i+1}\|}{|\det(D\Phi^{i+1}_{\xi_{0}})|}

The following proposition requires no assumptions except that the map $\Phi^{k}$ is $C^{2}$ at the point $\xi_{0}$ and that hyperbolic coordinates $e^{(k)},f^{(k)}$ are defined, which we now assume for the rest of this section.

Proposition 5.1.

For $\varsigma=x,y$ we have

\|Df^{(k)}\|\leq A_{k}\mathfrak{E}^{(k)}_{0}+A_{k}\sum_{i=1}^{k-1}\mathfrak{E}^{(k)}_{i}+B_{k}\sum_{i=0}^{k-1}\mathfrak{F}^{(k)}_{i}.

(61)

The remainder of this section is devoted to the proof of Proposition 5.1. First of all, in view of (4) and Remark 2.1, we define the linear map $\mathcal{L}^{(k)}_{\xi_{0}}:T_{\xi_{0}}M\to T_{\xi_{0}}M$ by:

\mathcal{L}^{(k)}_{\xi_{0}}=\mathcal{L}^{(k)}=(D\Phi^{k}_{\xi_{0}})^{*}\circ D\Phi_{\xi_{0}}^{k}

(62)

for which we have

\mathcal{L}^{(k)}e^{(k)}=\|e^{(k)}_{k}\|^{2}e^{(k)}\quad\textrm{and}\quad\mathcal{L}^{(k)}f^{(k)}=\|f^{(k)}_{k}\|^{2}f^{(k)}.

(63)

We omit explicit reference to $\xi_{0}$ in the interest of clarity. We will also write the components of the unit vector field $f^{(k)}$ as $f^{(k)}=(f^{(k)}_{1},f^{(k)}_{2})$ , and then write the covariant derivative of $f^{(k)}$ in $\mathbb{R}^{2}$ as

Df^{(k)}=\left(\partial_{x}f^{(k)},\partial_{y}f^{(k)}\right),

where

\partial_{x}f^{(k)}=\left(\partial_{x}f^{(k)}_{1},\partial_{x}f^{(k)}_{2}\right)\quad\textrm{and}\quad\partial_{y}f^{(k)}=\left(\partial_{y}f^{(k)}_{1},\partial_{y}f^{(k)}_{2}\right)

(64)

are the columns of $Df^{(k)}$ . We split the proof into three Lemmas.

Lemma 5.2.

\|Df^{(k)}\|\leq\sqrt{2}\max_{\varsigma=x,y}\left\{|\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle|\right\}.

(65)

Proof.

Notice that by Sublemma 4.5, putting $A=Df^{(k)}$ , and $\mathbf{a}_{1}=\partial_{x}f^{(k)}$ and $\mathbf{a}_{2}=\partial_{y}f^{(k)}$ , we have

\|Df^{(k)}\|\leq\sqrt{2}\max_{\varsigma=x,y}\left\{\partial_{\varsigma}f^{(k)}\right\}.

(66)

Additionally, by orthonormality of $\{e^{(k)},f^{(k)}\}$ , we obtain:

\partial_{\varsigma}f^{(k)}=\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle e^{(k)}+\langle f^{(k)},\partial_{\varsigma}f^{(k)}\rangle f^{(k)}.

(67)

Writing the columns of $Df^{(k)}$ as in (64), equations (66) and (67) together give us:

\|Df^{(k)}\|\leq\sqrt{2}\max_{\varsigma=x,y}\{\|\partial_{\varsigma}f^{(k)}\|\}\leq\sqrt{2}\max_{\varsigma=x,y}\{|\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle|+|\langle f^{(k)},\partial_{\varsigma}f^{(k)}\rangle|\}.

Differentiating the equality $\|f^{(k)}\|^{2}=\langle f^{(k)},f^{(k)}\rangle=1$ we get that $|\langle f^{(k)},\partial_{\varsigma}f^{(k)}\rangle|=0$ , thus obtaining (65). ∎

Lemma 5.3.

For $\varsigma=x,y$ ,

|\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle|\leq\frac{|\langle e^{(k)},(\partial_{\varsigma}\mathcal{L}^{(k)})f^{(k)}\rangle|}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}.

(68)

Proof.

By differentiating the second equation in (63), we have

(\partial_{\varsigma}\mathcal{L}^{(k)})f^{(k)}+\mathcal{L}^{(k)}\partial_{\varsigma}f^{(k)}=(\partial_{\varsigma}\|f^{(k)}_{k}\|^{2})f^{(k)}+\|f^{(k)}_{k}\|^{2}\partial_{\varsigma}f^{(k)}.

Taking the scalar product with $e^{(k)}$ , and using the fact that $e^{(k)},f^{(k)}$ are orthogonal, gives

\langle e^{(k)},(\partial_{\varsigma}\mathcal{L}^{(k)})f^{(k)}\rangle+\langle e^{(k)},\mathcal{L}^{(k)}\partial_{\varsigma}f^{(k)}\rangle=\langle e^{(k)},\|f^{(k)}_{k}\|^{2}\partial_{\varsigma}f^{(k)}\rangle.

(69)

Using the first equation in (63) and the fact that the matrix $\mathcal{L}^{(k)}=(D\Phi^{k}_{\xi_{0}})^{*}\circ(D\Phi^{k}_{\xi_{0}})$ is self-adjoint, we have

\langle e^{(k)},\mathcal{L}^{(k)}\partial_{\varsigma}f^{(k)}\rangle=\langle\mathcal{L}^{(k)}e^{(k)},\partial_{\varsigma}f^{(k)}\rangle=\|e^{(k)}_{k}\|^{2}\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle.

Substituting this last equality into (69) gives

\langle e^{(k)},(\partial_{\varsigma}\mathcal{L}^{(k)})f^{(k)}\rangle+\|e^{(k)}_{k}\|^{2}\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle=\|f^{(k)}_{k}\|^{2}\langle e^{(k)},\partial_{\varsigma}f^{(k)}\rangle,

which gives (68). ∎

Lemma 5.4.

For $\varsigma=x,y$ ,

|\langle e^{(k)},(\partial_{\varsigma}\mathcal{L}^{(k)})f^{(k)}\rangle|\leq\|f^{(k)}_{k}\|^{2}\sum_{i=0}^{k-1}\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e^{(k)}_{i}\|\|e^{(k)}_{i+1}\|}{|\det(D\Phi^{i+1}_{\xi_{0}})|}+\|e^{(k)}_{k}\|^{2}\sum_{i=0}^{k-1}\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\|\|f^{(k)}_{i+1}\|}{|\det(D\Phi^{i+1}_{\xi_{0}})|}.

(70)

Proof.

By the Leibniz product rule, we have:

\partial_{\varsigma}\mathcal{L}^{(k)}=\partial_{\varsigma}[(D\Phi^{k}_{\xi_{0}})^{*}(D\Phi^{k}_{\xi_{0}})]=[\partial_{\varsigma}(D\Phi^{k}_{\xi_{0}})]^{*}(D\Phi^{k}_{\xi_{0}})+(D\Phi^{k}_{\xi_{0}})^{*}[\partial_{\varsigma}(D\Phi^{k}_{\xi_{0}})].

(71)

By inductively applying the Leibniz rule to $\partial_{\varsigma}(D\Phi_{\xi_{0}}^{k})=\partial_{\varsigma}(D\Phi_{\xi_{k-1}}\cdots D\Phi_{\xi_{0}})$ , we obtain:

\partial_{\varsigma}(D\Phi_{\xi_{0}}^{k})=\sum_{i=0}^{k-1}(D\Phi_{\xi_{i+1}}^{k-i-1})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi_{\xi_{0}}^{i}).

(72)

By (71) and (72):

\displaystyle\partial_{\varsigma}\mathcal{L}^{(k)}

\displaystyle=\left(\sum_{i=0}^{k-1}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})\right)^{*}(D\Phi^{k}_{\xi_{0}})+(D\Phi^{k}_{\xi_{0}})^{*}\left(\sum_{i=0}^{k-1}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})\right).

Applying $\partial_{\varsigma}\mathcal{L}^{(k)}$ to $f^{(k)}$ and taking the scalar product with $e^{(k)}$ gives:

$\displaystyle\langle e^{(k)},\>(\partial_{\varsigma}\mathcal{L}^{(k)})f^{(k)}\rangle=$	$\displaystyle\bigg{\langle}e^{(k)},\>\left(\sum_{i=0}^{k-1}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})\right)^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}\bigg{\rangle}$	(73)
	$\displaystyle+\bigg{\langle}e^{(k)},\>(D\Phi^{k}_{\xi_{0}})^{*}\left(\sum_{i=0}^{k-1}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})\right)f^{k)}\bigg{\rangle}$
$\displaystyle=$	$\displaystyle\sum_{i=0}^{k-1}\bigg{\langle}e^{(k)},\>\left((D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})\right)^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}\bigg{\rangle}$
	$\displaystyle+\sum_{i=0}^{k-1}\bigg{\langle}e^{(k)},\>(D\Phi^{k}_{\xi_{0}})^{*}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})f^{k)}\bigg{\rangle}$
	$\displaystyle=\sum_{i=0}^{k-1}\mathfrak{A}_{i}+\sum_{i=0}^{k-1}\mathfrak{B}_{i},$

where:

	$\displaystyle\mathfrak{A}_{i}$	$\displaystyle=\bigg{\langle}e^{(k)},\>\left((D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})\right)^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}\bigg{\rangle},$
	$\displaystyle\mathfrak{B}_{i}$	$\displaystyle=\bigg{\langle}e^{(k)},\>(D\Phi^{k}_{\xi_{0}})^{*}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})f^{k)}\bigg{\rangle}.$

Consider the summands in the first sum of the right hand side of (73). We have:

$\displaystyle\mathfrak{A}_{i}$	$\displaystyle=\Big{\langle}(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})e^{(k)},\>D\Phi^{k}_{\xi_{0}}f^{(k)}\Big{\rangle}$	(74)
	$\displaystyle=\Big{\langle}[\partial_{\varsigma}D\Phi_{\xi_{i}}](D\Phi^{i}_{\xi_{0}})e^{(k)},\>(D\Phi^{k-i-1}_{\xi_{i+1}})^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}\Big{\rangle}$
	$\displaystyle=\Big{\langle}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>(D\Phi^{k-i-1}_{\xi_{i+1}})^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}\Big{\rangle}.$

Observing that $D\Phi_{\xi_{i+1}}^{k-i-1}=D\Phi_{\xi_{0}}^{k}(D\Phi_{\xi_{0}}^{i+1})^{-1}$ , we obtain:

(D\Phi^{k-i-1}_{\xi_{i+1}})^{*}=\left[(D\Phi^{i+1}_{\xi_{0}})^{-1}\right]^{*}(D\Phi^{k}_{\xi_{0}})^{*}.

(75)

Therefore, recalling that $\mathcal{L}^{(k)}f^{(k)}=(D\Phi_{\xi_{0}}^{k})^{*}(D\Phi_{\xi_{0}}^{k})f^{(k)}=\|f^{(k)}_{k}\|^{2}f^{(k)}$ , we obtain the expression:

\displaystyle(D\Phi^{k-i-1}_{\xi_{i+1}})^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}

\displaystyle=\big{[}(D\Phi^{i+1}_{\xi_{0}})^{-1}\big{]}^{*}(D\Phi^{k}_{\xi_{0}})^{*}(D\Phi^{k}_{\xi_{0}})f^{(k)}=\|f^{(k)}_{k}\|^{2}\big{[}(D\Phi^{i+1}_{\xi_{0}})^{-1}\big{]}^{*}f^{(k)}.

Applying this to the second argument of the right hand side of (74), we get:

\displaystyle\mathfrak{A}_{i}

\displaystyle=\|f^{(k)}_{k}\|^{2}\Big{\langle}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>\big{[}(D\Phi^{i+1}_{\xi_{0}})^{-1}\big{]}^{*}f^{(k)}\Big{\rangle}=\|f^{(k)}_{k}\|^{2}\Big{\langle}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>f^{(k)}\Big{\rangle}.

(76)

To estimate this inner product in (76), we use two basic properties of the determinant: given a $2\times 2$ matrix $A$ and two vectors $v$ , $w$ , we have:

\det(Av,Aw)=\det(A)\det(v,w)\quad\textrm{and}\quad\det(v,w)=\pm\langle v,w^{\top}\rangle

(77)

where $\det(v,w)$ refers to the determinant of a $2\times 2$ matrix whose columns are the vectors $v,w$ , and $w^{\top}$ is a vector orthogonal to $w$ with $\|w\|=\|w^{\top}\|$ . Taking $(f^{(k)})^{\top}=e^{(k)}$ , applying these properties to (76) gives us:

$\displaystyle\mathfrak{A}_{i}$	$\displaystyle=\\|f^{(k)}_{k}\\|^{2}\Big{\langle}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>f^{(k)}\Big{\rangle}$	(78)
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>(f^{(k)})^{\top}\Big{)}$
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>e^{(k)}\Big{)}$
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>(D\Phi^{i+1}_{\xi_{0}})^{-1}e^{(k)}_{i+1}\Big{)}$
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det(D\Phi_{\xi_{0}}^{i+1})^{-1}\det\Big{(}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>e^{(k)}_{i+1}\Big{)}.$

By (77) and the Cauchy-Schwarz inequality, we have

|\det(v,w)|=|\langle v,w^{\top}\rangle|\leq\|v\|\|w\|.

(79)

Taking the absolute value of both sides of (78) and applying (79) to $\det\big{(}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>e^{(k)}_{i+1}\big{)}$ , we obtain:

\displaystyle|\mathfrak{A}_{i}|\leq\|f^{(k)}_{k}\|^{2}\frac{\|\partial_{\varsigma}[D\Phi_{\xi_{i}}]e_{i}^{(k)}\|\|e^{(k)}_{i+1}\|}{|\det(D\Phi_{\xi_{0}}^{i+1})|}.

(80)

These are the summands in the first sum in (70). We perform a similar calculation on the summands of the second sum on the right hand side of (73). We have:

$\displaystyle\mathfrak{B}_{i}$	$\displaystyle=\Big{\langle}D\Phi^{k}_{\xi_{0}}e^{(k)},\>(D\Phi^{k-i-1}_{\xi_{i+1}})[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})f^{(k)}\Big{\rangle}$	(81)
	$\displaystyle=\Big{\langle}(D\Phi^{k-i-1}_{\xi_{i+1}})^{*}(D\Phi^{k}_{\xi_{0}})e^{(k)},\>[\partial_{\varsigma}(D\Phi_{\xi_{i}})](D\Phi^{i}_{\xi_{0}})f^{(k)}\Big{\rangle}$
	$\displaystyle=\Big{\langle}(D\Phi^{k-i-1}_{\xi_{i+1}})^{*}(D\Phi^{k}_{\xi_{0}})e^{(k)},\>[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{\rangle}.$

Recall now that $e^{(k)}$ is an eigenvector of $(D\Phi_{\xi_{0}}^{k})^{*}(D\Phi_{\xi_{0}}^{k})$ with eigenvalue $\|e^{(k)}_{k}\|^{2}$ . By (75), we get:

(D\Phi_{\xi_{i+1}}^{k-i-1})^{*}(D\Phi_{\xi_{0}}^{k})e^{(k)}=\big{[}(D\Phi_{\xi_{0}}^{i+1})^{-1}\big{]}^{*}(D\Phi_{\xi_{0}}^{k})^{*}(D\Phi_{\xi_{0}}^{k})e^{(k)}=\|e^{(k)}_{k}\|^{2}\big{[}(D\Phi_{\xi_{0}}^{i+1})^{-1}\big{]}^{*}e^{(k)}.

Applying this to the first argument of the right hand side of (81), we obtain:

	$\displaystyle\mathfrak{B}_{i}$	$\displaystyle=\\|e^{(k)}_{k}\\|^{2}\Big{\langle}\big{[}(D\Phi_{\xi_{0}}^{i+1})^{-1}\big{]}^{*}e^{(k)},\>[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{\rangle}$		(82)
		$\displaystyle=\\|e^{(k)}_{k}\\|^{2}\Big{\langle}e^{(k)},\>(D\Phi_{\xi_{0}}^{i+1})^{-1}[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{\rangle}.$		(82)

We again use (77) to estimate this inner product, this time taking $(e^{(k)})^{\top}=f^{(k)}$ :

$\displaystyle\mathfrak{B}_{i}$	$\displaystyle=\pm\\|e^{(k)}_{k}\\|^{2}\det\Big{(}f^{(k)},\>(D\Phi_{\xi_{0}}^{i+1})^{-1}[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{)}$	(83)
	$\displaystyle=\pm\\|e^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi_{\xi_{0}}^{i+1})^{-1}f^{(k)}_{i+1},\>(D\Phi_{\xi_{0}}^{i+1})^{-1}[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{)}$
	$\displaystyle=\pm\\|e^{(k)}_{k}\\|^{2}\det(D\Phi_{\xi_{0}}^{i+1})^{-1}\det\Big{(}f^{(k)}_{i+1},\>[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{)}.$

Using (79), taking the absolute value of both sides of (83) gives us:

\displaystyle|\mathfrak{B}_{i}|\leq\|e^{(k)}_{k}\|^{2}\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\|\|f^{(k)}_{i+1}\|}{|\det(D\Phi_{\xi_{0}}^{i+1}|}.

(84)

These are the summands of the second sum in (70). Now, (70) follows after taking the absolute value of (73) and applying (80) and (84). ∎

Proof of Proposition 5.1.

The proposition is an immediate consequence of Lemmas 5.2 - 5.4. ∎

6 Slow Variation for Quasi-Hyperbolic Points

We now consider the general bound (61) obtained in Proposition 5.1 and will use the quasi-hyperbolicity conditions to get more explicit bounds for the three terms on the right hand side of (61). First of all, in Section 6.1 we will prove the following.

Proposition 6.1.

For $\varsigma=x,y$ we have

\|Df^{(k)}\|\leq K_{1}\left(\mathfrak{E}_{0}^{(k)}+\sum_{i=1}^{k-1}\mathfrak{E}_{i}^{(k)}+\frac{\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}}\sum_{i=0}^{k-1}\mathfrak{F}_{i}^{(k)}\right).

(85)

In Section 6.2 and 6.3 we estimate the first two terms on the right hand side of (85) assuming quasi-hyperbolicity Conditions (I) and (II) respectively. Then in Section 6.4 we estimate the third term and finally, in Section 6.5, we combine these estimates to complete the proof of the first inequality in Theorem 2.5.

6.1 Proof of Proposition 6.1

Proposition 6.1 follows immediately from Proposition 5.1 and the following Lemma.

Lemma 6.2.

For every $k\geq 1$ we have

A_{k}\leq K_{1}\quad\text{ and }\quad B_{k}\leq\frac{\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}}K_{1}.

Proof.

By the definition of eccentricity and (9) we have ${\|e^{(k)}_{k}\|}/{\|f^{(k)}_{k}\|}=C_{\xi_{0},k}\leq Bc^{k},$ and therefore

\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}\geq(1-B^{2}c^{2k})\|f^{(k)}_{k}\|^{2}\geq(1-B^{2}c^{2})\|f^{(k)}_{k}\|^{2}

and so

\frac{\|f^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}\leq\frac{1}{1-B^{2}c^{2}}\quad\textrm{and}\quad\frac{\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}\leq\frac{\|e^{(k)}_{k}\|^{2}}{(1-B^{2}c^{2})\|f^{(k)}_{k}\|^{2}}.

Recalling the definitions of $A_{k}$ and $B_{k}$ , this implies

A_{k}:=\frac{\sqrt{2}\|f^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}\leq\frac{\sqrt{2}}{1-B^{2}c^{2}}\quad\text{ and }\quad B_{k}:=\frac{\sqrt{2}\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}-\|e^{(k)}_{k}\|^{2}}\leq\frac{\sqrt{2}\|e^{(k)}_{k}\|^{2}}{(1-B^{2}c^{2})\|f^{(k)}_{k}\|^{2}}

which gives the statement in the Lemma. ∎

6.2 Estimates for $\mathfrak{E}_{i}^{(k)}$ with Quasi-Hyperbolicity, Condition (I)

We assume throughout this subsection that $\xi_{0}$ is singular quasi-hyperbolic up to time $k$ and satisfies (I).

Lemma 6.3.

\mathfrak{E}_{0}^{(k)}\leq\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|+Q_{3}c

(86)

Proof.

By (45) and (46), as well as by (10) and (34):

	$\displaystyle\\|[\partial_{\varsigma}(D\Phi_{\xi_{0}})]e^{(k)}\\|$	$\displaystyle\leq\\|[\partial_{\varsigma}(D\Phi_{\xi_{0}})]e^{(1)}\\|+\\|\partial_{\varsigma}(D\Phi_{\xi_{0}})\\|\\|e^{(k)}-e^{(1)}\\|$
		$\displaystyle\leq\\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\\|+\\|D^{2}\Phi_{\xi_{0}}\\|\\|e^{(k)}-e^{(1)}\\|$
		$\displaystyle\leq\\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\\|+\frac{Q_{1}D\Gamma^{2}\widetilde{\Gamma}c}{\lambda}$

∎

Lemma 6.4.

\sum_{i=1}^{k-1}\mathfrak{E}_{i}^{(k)}\leq Q_{4}c.

(87)

Proof.

We observe that $\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})e^{(k)}_{i}\|\leq\|\partial_{\varsigma}(D\Phi_{\xi_{i}})\|\|e^{(k)}_{i}\|$ . By (10) and (45), we get:

\|\partial_{\varsigma}(D\Phi_{\xi_{i}})\|\leq\|D^{2}\Phi_{\xi_{i}}\|\leq D\Gamma\widetilde{\Gamma}^{i}.

(88)

From (35) and (88), we get:

\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e_{i}^{(k)}\|\leq Q_{1}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda}\right)^{i}D\Gamma\widetilde{\Gamma}^{i}=Q_{1}D\Gamma\left(\frac{\Gamma^{2}\widetilde{\Gamma}^{2}c}{\lambda}\right)^{i}.

(89)

By (36) and (89), we get:

\mathfrak{E}_{i}^{(k)}=\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e^{(k)}_{i}\|\|e^{(k)}_{i+1}\|}{|\det(D\Phi_{\xi_{0}}^{i+1})|}\leq Q_{1}D\Gamma\left(\frac{\Gamma^{2}\widetilde{\Gamma}^{2}c}{\lambda}\right)^{i}Q_{2}\left(\frac{\Gamma\widetilde{\Gamma}}{\lambda^{2}}\right)^{i+1}=\frac{Q_{1}Q_{2}D\Gamma^{2}\widetilde{\Gamma}}{\lambda^{2}}\left(\frac{\Gamma^{3}\widetilde{\Gamma}^{3}c}{\lambda^{3}}\right)^{i}.

(90)

Using the final inequality in (I):

\displaystyle\sum_{i=1}^{k-1}\mathfrak{E}_{i}^{(k)}\leq\frac{Q_{1}Q_{2}D\Gamma^{2}\widetilde{\Gamma}}{\lambda^{2}}\sum_{i=1}^{\infty}\left(\frac{\Gamma^{3}\widetilde{\Gamma}^{3}c}{\lambda^{3}}\right)^{i}=\frac{Q_{1}Q_{2}D\Gamma^{2}\widetilde{\Gamma}}{\lambda^{2}}\frac{1}{1-\frac{\Gamma^{3}\widetilde{\Gamma}^{3}c}{\lambda^{3}}}\frac{\Gamma^{3}\widetilde{\Gamma}^{3}c}{\lambda^{3}}=\frac{Q_{1}Q_{2}D\Gamma^{5}\widetilde{\Gamma}^{4}c}{\lambda^{2}(\lambda^{3}-\Gamma^{3}\widetilde{\Gamma}^{3}c)}.

∎

6.3 Estimates for $\mathfrak{E}_{i}^{(k)}$ with Quasi-Hyperbolicity, Condition (II)

We assume now that $\xi_{0}$ is quasi-hyperbolic up to time $k$ and satisfies (II).

Lemma 6.5.

\mathfrak{E}_{0}^{(k)}\leq\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|+\widetilde{Q}_{3}\frac{c}{\tilde{c}}.

(91)

Proof.

We note that by (45), (46), (10), and (40):

	$\displaystyle\\|[\partial_{\varsigma}(D\Phi_{\xi_{0}})]e^{(k)}\\|$	$\displaystyle\leq\\|[\partial_{\varsigma}(D\Phi_{\xi_{0}})]e^{(1)}\\|+\\|\partial_{\varsigma}(D\Phi_{\xi_{0}})\\|\\|e^{(k)}-e^{(1)}\\|$
		$\displaystyle\leq\\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\\|+\\|D^{2}\Phi_{\xi_{0}}\\|\\|e^{(k)}-e^{(1)}\\|$
		$\displaystyle\leq\\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\\|+\frac{\widetilde{Q}_{1}D\Gamma c}{\tilde{c}}.$

∎

Lemma 6.6.

\sum_{i=1}^{k-1}\mathfrak{E}_{i}^{(k)}\leq\widetilde{Q}_{4}\frac{c}{\tilde{c}}

(92)

Proof.

We observe that $\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e^{(k)}_{i}\|\leq\|\partial_{\varsigma}(D\Phi_{\xi_{i}})\|\|e^{(k)}_{i}\|$ . By (10) and (45), we get:

\|\partial_{\varsigma}(D\Phi_{\xi_{i}})\|\leq\|D^{2}\Phi_{\xi_{i}}\|\leq D\Gamma\widetilde{\Gamma}^{i}.

(93)

From (41) and (93), we get:

\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e_{i}^{(k)}\|\leq\widetilde{Q}_{1}\left(\frac{\Gamma c}{\tilde{c}}\right)^{i}D\Gamma\widetilde{\Gamma}^{i}=\widetilde{Q}_{1}D\Gamma\left(\frac{\Gamma\widetilde{\Gamma}c}{\tilde{c}}\right)^{i}.

(94)

By (42) and (94), we get:

\mathfrak{E}_{i}^{(k)}=\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]e^{(k)}_{i}\|\|e^{(k)}_{i+1}\|}{|\det(D\Phi_{\xi_{0}}^{i+1})|}\leq\widetilde{Q}_{1}D\Gamma\left(\frac{\Gamma\widetilde{\Gamma}c}{\tilde{c}}\right)^{i}\widetilde{Q}_{2}\left(\frac{\Gamma}{\lambda^{2}\tilde{c}}\right)^{i+1}=\frac{\widetilde{Q}_{1}\widetilde{Q}_{2}D\Gamma^{2}}{\lambda^{2}\tilde{c}}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda^{2}\tilde{c}^{2}}\right)^{i}.

(95)

Summing up these terms gives us:

\displaystyle\sum_{i=1}^{k-1}\mathfrak{E}_{i}^{(k)}\leq\frac{\widetilde{Q}_{1}\widetilde{Q}_{2}D\Gamma^{2}}{\lambda^{2}\tilde{c}}\sum_{i=1}^{\infty}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda^{2}\tilde{c}^{2}}\right)^{i}=\frac{\widetilde{Q}_{1}\widetilde{Q}_{2}D\Gamma^{2}}{\lambda^{2}\tilde{c}}\frac{1}{1-\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda^{2}\tilde{c}^{2}}}\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda^{2}\tilde{c}^{2}}=\frac{\widetilde{Q}_{1}\widetilde{Q}_{2}D\Gamma^{4}\widetilde{\Gamma}c}{\lambda^{2}\tilde{c}(\lambda^{2}\tilde{c}^{2}-\Gamma^{2}\widetilde{\Gamma}c)}.

∎

6.4 Estimates for $\mathfrak{F}_{i}^{(k)}$

We now estimate the third term of (61). Unlike the estimates for the $\mathfrak{E}_{i}^{(k)}$ terms, the $\mathfrak{F}_{i}^{(k)}$ terms do not require separate assumptions depending on whether $\xi_{0}$ satisfies conditions (I) or (II).

Lemma 6.7.

\frac{\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}}\sum_{i=0}^{k-1}\mathfrak{F}_{i}^{k-1}\leq Qc.

(96)

Proof.

First of all, we observe that

|\det(D\Phi_{\xi_{0}}^{i+1})|=\frac{|\det(D\Phi_{\xi_{0}}^{k})|}{|\det(D\Phi_{\xi_{i+1}}^{k-i-1})|}=\frac{\|e^{(k)}_{k}\|\|D\Phi_{\xi_{0}}^{k}\|}{|\det(D\Phi_{\xi_{0}}^{k-i-1})|},

(97)

and therefore, using also the facts that $\|f^{(k)}_{i+1}\|\leq\|D\Phi^{i+1}_{\xi_{0}}\|$ and $\|\partial_{\varsigma}(D\Phi_{\xi_{i}})\|\leq\|D^{2}\Phi_{\xi_{i}}\|$ ,

	$\displaystyle\frac{\\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\\|\\|f^{(k)}_{i+1}\\|}{\|\det(D\Phi_{\xi_{0}}^{i+1})\|}$	$\displaystyle\leq\frac{\\|D^{2}\Phi_{\xi_{i}}\\|\\|D\Phi^{i}_{\xi_{0}}\\|\\|D\Phi^{i+1}_{\xi_{0}}\\|}{\|\det(D\Phi_{\xi_{0}}^{i+1})\|}$		(98)
		$\displaystyle=\frac{\\|D^{2}\Phi_{\xi_{i}}\\|\\|D\Phi_{\xi_{0}}^{i}\\|\\|D\Phi_{\xi_{0}}^{i+1}\\|\|\det(D\Phi_{\xi_{i+1}}^{k-i-1})\|}{\\|e^{(k)}_{k}\\|\\|D\Phi_{\xi_{0}}^{k}\\|}.$		(98)

Therefore, applying (9) and (10) from Definition 2.3 to (98), we obtain:

\displaystyle\mathfrak{F}_{i}^{(k)}

\displaystyle=\frac{\|[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\|\|f^{(k)}_{i+1}\|}{|\det(D\Phi_{\xi_{0}}^{i+1})|}\leq\frac{D^{3}\Gamma^{2i+2}\widetilde{\Gamma}^{i}b^{k-i-1}}{\|e^{(k)}_{k}\|C\lambda^{k}}=\frac{D^{3}\Gamma^{2}b^{k-1}}{\|e^{(k)}_{k}\|C\lambda^{k}}\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{i},

(99)

and therefore,

\frac{\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}}\sum_{i=0}^{k-1}\mathfrak{F}_{i}^{k-1}\leq\frac{\|e^{(k)}_{k}\|D^{3}\Gamma^{2}b^{k-1}}{\|f^{(k)}_{k}\|^{2}C\lambda^{k}}\sum_{i=0}^{k-1}\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{i}

(100)

Estimating this sum, we find:

\sum_{i=0}^{k-1}\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{i}=\frac{\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{k}-1}{\frac{\Gamma^{2}\widetilde{\Gamma}}{b}-1}\leq\frac{\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{k}}{\frac{\Gamma^{2}\widetilde{\Gamma}}{b}-1}=\frac{b}{\Gamma^{2}\widetilde{\Gamma}-b}\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{k}

(101)

(note the inequality above is where we use the final assumption in (11) that $\Gamma^{2}\widetilde{\Gamma}>b$ ). Plugging this into (100), and also recalling that $\|e^{(k)}_{k}\|/\|f^{(k)}_{k}\|<Bc^{k}$ and $\|f^{(k)}_{k}\|\geq C\lambda^{k}$ , we have:

	$\displaystyle\frac{\\|e^{(k)}_{k}\\|^{2}}{\\|f^{(k)}_{k}\\|^{2}}\sum_{i=0}^{k-1}\mathfrak{F}_{i}^{k-1}$	$\displaystyle\leq\frac{\\|e^{(k)}_{k}\\|D^{3}\Gamma^{2}b^{k}}{\\|f^{(k)}_{k}\\|^{2}C\lambda^{k}(\Gamma^{2}\widetilde{\Gamma}-b)}\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{k}$
		$\displaystyle\leq\frac{BD^{3}\Gamma^{2}(bc)^{k}}{C^{2}\lambda^{2k}(\Gamma^{2}\widetilde{\Gamma}-b)}\left(\frac{\Gamma^{2}\widetilde{\Gamma}}{b}\right)^{k}$
		$\displaystyle=\frac{BD^{3}\Gamma^{2}}{C^{2}(\Gamma^{2}\widetilde{\Gamma}-b)}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda^{2}}\right)^{k}.$

Finally, since $\widetilde{\Gamma}\geq 1$ and $\Gamma>\lambda$ , and since $\tilde{c}<1$ , both (I) and (II) imply that $\Gamma^{2}\widetilde{\Gamma}c/\lambda^{2}\leq 1$ . Therefore,

\frac{\|e^{(k)}_{k}\|^{2}}{\|f^{(k)}_{k}\|^{2}}\sum_{i=0}^{k-1}\mathfrak{F}_{i}^{k-1}\leq\frac{BD^{3}\Gamma^{2}}{C^{2}(\Gamma^{2}\widetilde{\Gamma}-b)}\left(\frac{\Gamma^{2}\widetilde{\Gamma}c}{\lambda^{2}}\right)^{k}\leq\frac{BD^{3}\Gamma^{2}}{C^{2}(\Gamma^{2}\widetilde{\Gamma}-b)}\frac{\Gamma^{2}\widetilde{\Gamma}}{\lambda^{2}}c

which gives the statement in the Lemma. ∎

6.5 Proof of first inequality in Theorem 2.5

Suppose condition (I) is satisfied. Substituting the bounds in Lemmas 6.3, 6.4, and 6.7 into (85) we get

\|Df^{(k)}\|\leq K_{1}(\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|+Q_{3}c+Q_{4}c+Qc)

which gives the first inequality in Theorem 2.5 with $K_{2}=K_{1}(Q_{3}+Q_{4}+Q_{5})$ . Similarly, supposing condition (II) is satisfied, substituting the bounds in 6.5, 6.6, and 6.7, into (85) we get

\|Df^{(k)}\|\leq K_{1}(\|D^{2}\Phi_{\xi_{0}}(e^{(1)},\cdot)\|+\widetilde{Q}_{3}c+\widetilde{Q}_{4}c+Qc)

which gives the first inequality in Theorem 2.5 with $K_{2}=K_{1}(\widetilde{Q}_{3}+\widetilde{Q}_{4}+Q_{5})$ .

References

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[2] Luis Barreira and Yakov Pesin “Nonuniform hyperbolicity” Dynamics of systems with nonzero Lyapunov exponents 115, Encyclopedia of Mathematics and its Applications Cambridge University Press, Cambridge, 2007, pp. xiv+513 URL: http://dx.doi.org/10.1017/CBO9781107326026
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[5] Katie Bloor and Stefano Luzzatto “Some remarks on the geometry of the standard map” In International Journal of Bifurcation and Chaos 19.7, 2009, pp. 2213–2232 DOI: 10.1142/S0218127409024025
[6] Rufus Bowen “Equilibrium states and the ergodic theory of Anosov diffeomorphisms”, Lecture Notes in Mathematics, Vol. 470 Berlin: Springer-Verlag, 1975, pp. i+108
[7] Rufus Bowen and David Ruelle “The ergodic theory of Axiom A flows” In Invent. Math. 29.3, 1975, pp. 181–202
[8] Vaughn Climenhaga, Stefano Luzzatto and Yakov Pesin “SRB measures and Young towers for surface diffeomorphisms” In Annales Henri Poincaré 23, 2023 URL: https://link.springer.com/article/10.1007/s00023-021-01113-5
[9] A Giorgilli and V F Lazutkin “Some remarks on the problem of ergodicity of the standard map” In Physics Letters. A 272.5-6, 2000, pp. 359–367
[10] Franz Hofbauer and Gerhard Keller “Quadratic maps without asymptotic measure” In Comm. Math. Phys. 127.2, 1990, pp. 319–337 URL: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-127/issue-2/Quadratic-maps-without-asymptotic-measure/cmp/1104180141.full
[11] Mark Holland and Stefano Luzzatto “A new proof of the stable manifold theorem for hyperbolic fixed points on surfaces” In Journal of Difference Equations and Applications 11.6, 2005, pp. 535–551 URL: http://dx.doi.org/10.1080/10236190500127554
[12] Mark Holland and Stefano Luzzatto “Stable manifolds under very weak hyperbolicity conditions” In Journal of Differential Equations 221.2, 2006, pp. 444–469 DOI: 10.1016/j.jde.2005.07.013
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[22] Marcelo Viana “Multidimensional nonhyperbolic attractors” In Inst. Hautes Études Sci. Publ. Math. 85, 1997, pp. 63–96 URL: http://www.numdam.org/item?id=PMIHES_1997__85__63_0

Stefano Luzzatto
Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy.
https://www.stefanoluzzatto.net
luzzatto@ictp.it

Dominic Veconi
Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy.
Current Affiliation: Wake Forest University, Winston-Salem, NC, USA
https://dominic.veconi.com
veconid@wfu.edu

Khadim War
Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil
https://sites.google.com/view/khadim-war/home
warkhadim@gmail.com

$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\widetilde{C}_{\xi_{0},k}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|};$	(20)
$\displaystyle\\|e^{(k)}_{i}\\|$	$\displaystyle\leq\\|(D\Phi_{\xi_{0}}^{i})^{-1}\\|^{-1}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|};$	(21)
$\displaystyle\frac{\\|e^{(k)}_{i}\\|}{\|\det D\Phi_{\xi_{0}}^{i}\|}$	$\displaystyle\leq\frac{1}{\\|D\Phi_{\xi_{0}}^{i}\\|}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{\|\det D\Phi_{\xi_{i}}^{j-i}\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{0}}^{j+1}\\|}.$	(22)

$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\mathcal{T}_{\xi_{0},i}^{(k)}\widetilde{C}_{\xi_{0},k};$	(24)
$\displaystyle\\|e^{(k)}_{i}\\|$	$\displaystyle\leq\\|(D\Phi_{\xi_{0}}^{i})^{-1}\\|^{-1}+\\|D\Phi_{\xi_{0}}^{i}\\|\mathcal{T}_{\xi_{0},i}^{(k)}\widetilde{C}_{\xi_{0},k};$	(25)
$\displaystyle\frac{\\|e^{(k)}_{i}\\|}{\|\det D\Phi_{\xi_{0}}^{i}\|}$	$\displaystyle\leq\frac{1}{\\|D\Phi_{\xi_{0}}^{i}\\|}+\widetilde{C}_{\xi_{0},k}\\|D\Phi_{\xi_{0}}^{i}\\|\sum_{j=i}^{k-1}\frac{\|\det D\Phi_{\xi_{i}}^{j-i}\|}{\\|D\Phi_{\xi_{0}}^{j}\\|^{2}C_{\xi_{j},1}}.$	(26)

	$\displaystyle\\|e^{(k)}-e^{(i)}\\|$	$\displaystyle\leq\sum_{j=i}^{k-1}\\|e^{(j)}-e^{(j+1)}\\|$
		$\displaystyle\leq\sqrt{\frac{2}{1-C_{\xi_{0},i}^{2}}}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|}$
		$\displaystyle\leq\widetilde{C}_{\xi_{0},k}\sum_{j=i}^{k-1}\frac{C_{\xi_{0},j}\\|D\Phi_{\xi_{0}}^{j}\\|\\|D\Phi_{\xi_{j}}\\|}{\\|D\Phi_{\xi_{0}}^{j+1}\\|}.$

$\displaystyle\mathfrak{A}_{i}$	$\displaystyle=\\|f^{(k)}_{k}\\|^{2}\Big{\langle}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>f^{(k)}\Big{\rangle}$	(78)
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>(f^{(k)})^{\top}\Big{)}$
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>e^{(k)}\Big{)}$
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi^{i+1}_{\xi_{0}})^{-1}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>(D\Phi^{i+1}_{\xi_{0}})^{-1}e^{(k)}_{i+1}\Big{)}$
	$\displaystyle=\pm\\|f^{(k)}_{k}\\|^{2}\det(D\Phi_{\xi_{0}}^{i+1})^{-1}\det\Big{(}[\partial_{\varsigma}D\Phi_{\xi_{i}}]e^{(k)}_{i},\>e^{(k)}_{i+1}\Big{)}.$

$\displaystyle\mathfrak{B}_{i}$	$\displaystyle=\pm\\|e^{(k)}_{k}\\|^{2}\det\Big{(}f^{(k)},\>(D\Phi_{\xi_{0}}^{i+1})^{-1}[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{)}$	(83)
	$\displaystyle=\pm\\|e^{(k)}_{k}\\|^{2}\det\Big{(}(D\Phi_{\xi_{0}}^{i+1})^{-1}f^{(k)}_{i+1},\>(D\Phi_{\xi_{0}}^{i+1})^{-1}[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{)}$
	$\displaystyle=\pm\\|e^{(k)}_{k}\\|^{2}\det(D\Phi_{\xi_{0}}^{i+1})^{-1}\det\Big{(}f^{(k)}_{i+1},\>[\partial_{\varsigma}(D\Phi_{\xi_{i}})]f^{(k)}_{i}\Big{)}.$