^†^†thanks: I am thankful to C. Pugh and M. Hirsch for helpful and instructive discussions

Homoclinic tangencies in ${\mathbb{R}}^{n}$

Victoria Rayskin Victoria Rayskin Department of Mathematics
520 Portola Plaza
Box 951555
University of California
Los Angeles, CA 90095-1555, USA vrayskin@math.ucla.edu

Abstract.

Let $f:M\rightarrow M$ denote a diffeomorphism of a smooth manifold $M$ . Let $p\in M$ be its hyperbolic fixed point with stable and unstable manifolds $W_{S}$ and $W_{U}$ respectively. Assume that $W_{S}$ is a curve. Suppose that $W_{U}$ and $W_{S}$ have a degenerate homoclinic crossing at a point $B\neq p$ , i.e., they cross at $B$ tangentially with a finite order of contact.

It is shown that, subject to $C^{1}$ -linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$ . This proves the existence of a horseshoe structure arbitrarily close to $B$ , and extends a similar planar result of Homburg and Weiss [10].

Key words and phrases:

Homoclinic tangency, invariant manifolds,

\lambda

-Lemma, order of contact, horseshoe structure

1991 Mathematics Subject Classification:

37B10, 37C05, 37C15, 37D10

1. Introduction

The celebrated Theorem of Birkhoff-Smale (see, for instance, [17]) provides a method for rigorously concluding the existence of a horseshoe structure. The basic assumption of this theorem is the presence of a transverse homoclinic point.

Theorem 1.1 (Birkhoff-Smale).

Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be a diffeomorphism with a hyperbolic fixed point $p$ . If $q\not=p$ is a transverse homoclinic point, then there exist a hyperbolic invariant set $\Lambda$ , a set of bi-infinite sequences ${\Sigma}$ and a shift map $\sigma$ such that for some $m$ the following diagram commutes:

\begin{array}[]{ccc}\Lambda&\stackrel{{\scriptstyle f^{m}}}{{\rightarrow}}&\Lambda\\ h\downarrow&&\ \ \downarrow h\\ \Sigma&\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}&\Sigma\end{array}

Here $\Sigma=\{...A_{-2}A_{-1}.A_{0}A_{1}A_{2}...\}$ , $A_{i}\in\{0,...,N\}$ ,
$\sigma\{...A_{-2}A_{-1}.A_{0}A_{1}A_{2}...\}=\{...A_{-2}A_{-1}A_{0}.A_{1}A_{2}...\}$ and $h$ is a homeomorphism mapping $\Lambda$ onto $\Sigma$ .

The assumption of transversality is not easy to verify for a concrete dynamical system. There were many attempts to remove this assumption. (See K. Burns and H. Weiss [3], A. J. Homburg and H. Weiss [10], M. Hirsch [9], R. Churchill and D. Rod [4], Gavrilov and Shilnikov [5], [6], Gonchenko and Shilnikov [7]). These papers address the question of existence of horseshoes arbitrarily close to a homoclinic point in planar dynamics.

The object of the papers of Hirsch [9] and Churchill and Rod [4] was to show that, in the planar case, the manifolds $W_{S}$ and $W_{U}$ cross transversely at some other point, arbitrarily close to the original crossing, if the original crossing is two-sided (Figure 5, case IV) and of finite contact.

Churchill and Rod [4] imposed the additional assumptions that $f$ is analytic and area-preserving. In that case, by Moser’s Theorem [13], this diffeomorphism can be assumed to be in the Birkhoff normal form, and any branch of the stable (unstable) manifold can be re-parameterized so that $f^{n}(W_{U})\cap W_{S}$ contains a positive orbit $q^{\prime},fq^{\prime},f^{2}q^{\prime},...,f^{k}q^{\prime}$ that lies on a hyperbola. Then, direct calculations imply that $q^{\prime}$ is a transverse homoclinic point. But area preservation is a strong restriction. Furthermore, analyticity allows one to consider the case of a finite order of contact only.

Hirsch [9], replaced the above assumptions with the smooth linearizability assumption (near $p$ ). If we substitute $L$ , the linear part of $f$ , for $f$ , the order of contact will not be changed, and we can calculate the “slope” between two manifolds at the point $q^{\prime}$ , where $L^{n}(W_{U})$ crosses $W_{S}$ . If the crossing at $q^{\prime}$ is transverse, then $f^{n}(W_{U})=\Phi{L}^{n}{\Phi}^{-1}(W_{U})$ also crosses $W_{S}$ non-degenerately, where $\Phi$ is the $C^{1}$ linearizing map.

Homburg and Weiss [10] considered a surface diffeomorphism and its homoclinic one-sided (Figures 5, 4) tangency of arbitrary high (possibly infinite) order of contact. They proved that some power of this diffeomorphism has the full shift on two symbols as a topological factor. Then, Katok’s theorem [11], [12] implies that the map possesses a horseshoe arbitrarily close to homoclinic tangency.

Important cases of homoclinic tangencies were also described in works of Shilnikov, Gonchenko and Gavrilov (see [7], [5], [6]). They considered one-sided planar homoclinic tangencies.

Gavrilov and Shilnikov [5], [6] established a topological conjugacy (on a closed invariant set) between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic (one-sided) tangencies (Figures 5, 4) and the full shift on two symbols.

Gonchenko and Shilnikov considered higher orders of contact. Homburg and Weiss write: “Gonchenko claims that a proof of this result for finite order tangencies appeared in his unpublished (Russian) thesis in 1984. The result for finite order tangencies was announced by Gonchenko and Shilnikov [7] in 1986, but we are unable to find any proof in the literature.” We were unable to find such a proof in the existing literature either.

Homburg and Weiss, in their paper [10], state an Open Problem, asking whether their results could be extended to dimensions higher than two. In the present paper, we consider a diffeomorphism $f$ of a smooth manifold $M$ (of an arbitrary dimension) with a hyperbolic fixed point. We assume that one of the invariant manifolds of $f$ is a curve, that this curve has finite tangential contact with the other invariant manifold, and that the contact is not of Newhouse type (see Definition 3.1 and Figure 4). We prove (Theorem 3.2) that, subject to $C^{1}$ -linearizability of $f$ and the existence of a graph portion of the invariant manifold, a transverse homoclinic crossing will arise arbitrarily close to the tangential crossing. This implies the existence of a horseshoe arbitrarily close to the tangential crossing.

To prove this result, we consider (Section 3) tangent subspaces of stable and unstable manifolds at a certain sequence of their intersection points. We show that, at some intersection points, a pair of the tangent subspaces spans the entire ${\mathbb{R}}^{n}$ . In Section 2 we present basic definitions and lemmas that are used in Section 3. In particular, we prove tangential $\lambda$ -Lemma 2.14, essential for the proof of the main result. Also, in Section 2, we present a new definition of the order of contact of two manifolds. This definition does not require high differentiability and plays a key role in our approach.

2. Definitions and Preliminaries

In this section we give basic definitions and state several preliminary lemmas to be used in the proof of the main result. The references for this Section are V. I. Arnold, S. M. Guseĭn-Zade [1] and A. N. Varchenko, K. Burns and H. Weiss [3], M. Hirsch [9], A. J. Homburg and H. Weiss [10], S. Newhouse and J. Palis [14], J. Palis [15], V. Rayskin [16], R. Uribe-Vargas [18].

Throughout this section, we will be considering two submanifolds in a certain ambient smooth manifold (of arbitrary dimension). Suppose two such submanifolds meet at an isolated point $A$ . We will discuss the contact properties of these submanifolds at point $A$ . Since these properties are local, we may assume without loss of generality that the ambient manifold is just ${\mathbb{R}}^{n}$ .

First, assume that each manifold is a curve. The order of contact of two smooth curves has been defined and studied in works of Arnold, Guseĭn-Zade [1] and A. N. Varchenko, Burns and Weiss [3], Hirsch [9], Homburg and Weiss [10], Uribe-Vargas [18]. Their definitions of the $l$ -th order of contact require high differentiability; i.e. $l$ is the maximal integer, such that the first $l-1$ derivatives of the two curves coincide at the point of their contact. Our definition below does not require high differentiability and thus, allows us to apply $C^{1}$ -linearization in our main Theorem 3.2.

Definition 2.1.

Let ${\gamma}_{i}$ ( $i=1,2$ ) denote two immersed $C^{1}$ -curves in ${\mathbb{R}}^{n}$ . Suppose the two curves meet at an isolated point $A$ . Then we will say that a number $l$ is the order of contact of the curve $\gamma_{1}$ with the curve $\gamma_{2}$ at the point $A$ if there exist positive real numbers $m$ and $M$ such that for all points $x$ on $\gamma_{1}$ sufficiently close to $A$

m\leq\frac{d(x,\gamma_{2})}{\|x-A\|^{l}}\leq M\;.

Proposition 2.2.

If the curves are $C^{1}$ -smooth, then the definition of the order of contact is symmetric with respect to $\gamma_{1}$ and $\gamma_{2}$ .

Proof.

Since the curves are $C^{1}$ -smooth, we can assume that $\gamma_{2}$ is the $X$ -axis (in some coordinate system), and $\gamma_{1}$ is a graph of a $C^{1}$ function $f$ in this system. Let the order of contact of $\gamma_{1}$ with $\gamma_{2}$ be $l$ . By Definition 2.1 it means that

m\leq\frac{\|f(x)\|}{\left(x^{2}+\|f(x)\|^{2}\right)^{l/2}}\leq M\;,

so that there exists a constant $C$ such that for all sufficiently small $x$

m\leq\frac{\|f(x)\|}{|x|^{l}}\leq C\;,

which means that $\gamma_{1}$ is sandwiched between two order $l$ paraboloids ${\mathcal{P}}_{m}$ and ${\mathcal{P}}_{C}$ , respectively.

Let us now look at the order of contact of $\gamma_{2}$ with $\gamma_{1}$ . By the above, for any point $x\in X$

d(x,{\mathcal{P}}_{m})\leq d(x,\gamma_{1})\leq d(x,{\mathcal{P}}_{C})\;.

Since ${\mathcal{P}}_{m}$ and ${\mathcal{P}}_{C}$ are order $l$ paraboloids, the distances to them are uniformly equivalent to $|x|^{l}$ , whence the claim ∎

Remark 2.3.

Our notion of the order of contact is more general than the classical one. Obviously, if the order of contact of two curves is $l$ in the classical sense, then it is also $l$ in our sense. The converse, however, need not be true.

Remark 2.4.

Without the $C^{1}$ assumption Proposition 2.2 may fail. For instance, let us consider the graphs of the functions $\gamma_{1}(x)=0$ and $\gamma_{2}(x)=x^{2}+(1+\sin(1/x))x$ on the plane. $\gamma_{2}$ oscillates between $x^{2}$ and $x$ with the distance between consecutive crests of order $x^{2}$ . Then, the order of contact of $\gamma_{1}$ with $\gamma_{2}$ in the sense of our Definition 2.1 is 2, whereas the order of contact of $\gamma_{2}$ with $\gamma_{1}$ does not exists.

Naturally, the $l$ -th order of contact (in the sense of [1], [3], [9], [10], [18]) between two $C^{l}$ -curves is preserved under a $C^{1}$ -diffeomorphism. Lemma 2.5 below shows that the order of contact in the sense of our Definition 2.1 is preserved by $C^{1}$ -diffeomorphisms. In particular, our order of contact from Definition 2.1 does not depend on the choice of a coordinate chart in the ambient manifold, so that it is well-defined in the general manifold setup as well.

Lemma 2.5.

Given two $C^{1}$ -curves in ${\mathbb{R}}^{n}$ intersecting at an isolated point, any diffeomorphism of a neighborhood of this point preserves the order of contact of these curves.

Proof.

Let curves $\gamma_{1}$ and $\gamma_{2}$ have order of contact $l$ at some point $A$ . Then there are positive constants $m$ and $M$ such that for all points $x\in\gamma_{1}$ sufficiently close to $A$

m\leq\frac{\|x-y\|}{\|x-A\|^{l}}\leq M,

where $y=y(x)$ denotes a point on $\gamma_{2}$ minimizing the distance from $x$ to $\gamma_{2}$ . By the $C^{1}$ Mean Value Theorem applied to the diffeomorphism $\phi$ ,

\phi(x)-\phi(y)=\Bigg{[}\int_{0}^{1}(D\phi)_{\sigma(s)}ds\Bigg{]}(x-y),

where $\sigma:[0,1]\to{\mathbb{R}}^{n}$ is a path connecting the points $x=\sigma(0)$ and $y=\sigma(1)$ . As $x\to A$ , $y(x)\to A$ . Then, $\sigma(s)\to A$ , and the matrix $\int_{0}^{1}(D\phi)_{\sigma(s)}ds$ tends to the invertible matrix $(D\phi)_{A}$ , hence the claim. ∎

Let us now pass to a discussion of the order of contact for higher dimensional intersecting manifolds.

Definition 2.6.

Let $S_{1}$ and $S_{2}$ denote two immersed $C^{1}$ -manifolds in ${\mathbb{R}}^{n}$ . Suppose the two manifolds meet at an isolated point $A$ . Their order of contact $l$ at $A$ is the supremum of the orders of contact of curves $\gamma_{1}\subset S_{1}$ and $\gamma_{2}\subset S_{2}$ passing through point $A$ .

As it follows from Lemma 2.5, the order of contact is preserved under $C^{1}$ -diffeomorphisms.

The well known ${\lambda}$ -Lemma of Palis [15] gives an important description of chaotic dynamics. The basic assumption of this theorem is the presence of a transverse homoclinic point, which means that the manifolds have contact of order 1.

Theorem 2.7 ( $\lambda$ -Lemma, Palis).

Let $f$ be a diffeomorphism of ${\mathbb{R}}^{n}$ with a hyperbolic fixed point at $0$ and $m$ - and $p$ -dimensional stable and unstable manifolds $W_{S}$ and $W_{U}$ ( $m+p=n$ ). Let $D$ be a $p$ -disk in $W_{U}$ , and $w$ be another $p$ -disk in $W_{U}$ meeting $W_{S}$ at some point $A$ transversely. Then $\bigcup_{n\geq 0}f^{n}(w)$ contains $p$ -disks arbitrarily $C^{1}$ -close to $D$ .

Later in this section we will prove an analog of this lemma (Lemma 2.14) for non-transversal homoclinic intersections.

For the estimates in the proofs of the Singular $\lambda$ -Lemma (i.e., in the case when the order of contact is greater than 1) and the Main Theorem we need the following definitions. The first one, the definition of a graph portion, deals with the shape of a global invariant manifold in the vicinity of a homoclinic point. This homoclinic point must be chosen close to a hyperbolic reference point. Then, the local properties of the invariant manifold are checked. The second definition describes a group of diffeomorphisms that can be locally represented (after a $C^{1}$ change of coordinates) in some special normal form.

Definition 2.8.

Let $f$ be a diffeomorphism of ${\mathbb{R}}^{n}$ with a hyperbolic fixed point at the origin and $U\in$ ${\mathbb{R}}^{n}$ be some small neighborhood of $0$ . Denote by $W_{S}$ (resp., $W_{U}$ ) the associated stable (resp., unstable) manifold, and by $m$ (resp., $p$ ) its dimension ( $m+p=n$ ). Let $A$ be a homoclinic point of $W_{S}$ and $W_{U}$ . Denote by $\mathcal{V}$ a small $p$ -neighborhood in $W_{U}$ around the origin. Define a local coordinate system $E_{1}$ at $0$ , which spans $\mathcal{V}$ . Similarly, define a local coordinate system $E_{2}$ which spans $W_{S}$ near $0$ . Let $E=E_{1}+E_{2}$ , $E\subset U$ . A $p$ -neighborhood $\Lambda\subset W_{U}$ is a graph portion in $U$ , associated with the homoclinic point $A$ , if for some $n>0$ $f^{n}(A)\in\Lambda$ , $\Lambda\subset(U\cap W_{U})$ , and $\Lambda$ is a graph in $E$ -coordinates of some $C^{1}$ -function defined on $\mathcal{V}$ .

\psfrag{L}{$\Lambda$}\psfrag{A}{$A$}\psfrag{0}{$0$}\psfrag{V}{$\mathcal{V}$}\psfrag{WS}{$W_{S}$}\psfrag{WU}{$W_{U}$}\epsfbox{fig_graph.eps}

Figure 1. In this figure,

\Lambda

is not a graph portion of the manifold

W_{U}

, because, for any

n

, an iteration of

\Lambda

with diffeomorphism

f^{n}

is never a graph in the given coordinates.

Definition 2.9.

We will say that diffeomorphism $f$ with a hyperbolic fixed point $0$ satisfies the normal form condition, if there exists a local $C^{1}$ change of coordinates in a neighbourhood of $0$ , under which $f$ takes the following normal form: $f(x,y)=(S_{1}(x,y),S_{2}(x,y))$ , where $S_{1}$ and $S_{2}$ belong to expanding and contracting subspaces respectively, and $S_{2}$ is linear, i.e $S_{2}(x,y)={\mathcal{B}}y$ , with $\|{\mathcal{B}}\|<1$ .

We are interested in the class of diffeomorphisms satisfying the normal form condition described above. The next lemma (Lemma 2.12) allows us to ascertain belonging to this class by verifying certain conditions (called resonances) on the eigenvalues of the linear part of the map. The following two definitions are used in the lemma.

Definition 2.10.

Let $f$ be a diffeomorphism of ${\mathbb{R}}^{n}$ with the linear part

(({\mathcal{A}}x)_{1},\dots,({\mathcal{A}}x)_{p},({\mathcal{B}}y)_{1},\dots,({\mathcal{B}}y)_{m})

at a hyperbolic fixed point $0$ , where $m$ and $p$ are the dimensions of its stable and unstable manifolds, respectively. Then, $f$ satisfies the mixed second order resonance condition if there exist $a\in\mathop{\rm spec}{\mathcal{A}}$ and $b\in\mathop{\rm spec}{\mathcal{B}}$ with $ab\in(\mathop{\rm spec}{\mathcal{A}}\cup\mathop{\rm spec}{\mathcal{B}})$ .

Definition 2.11.

We will say that $f$ has mixed second order resonance in its contracting coordinates if there exist $a\in\mathop{\rm spec}{\mathcal{A}}$ , $b\in\mathop{\rm spec}{\mathcal{B}}$ with $ab\in\mathop{\rm spec}{\mathcal{B}}$ .

Lemma 2.12.

If a $C^{\infty}$ diffeomorphism $f$ with a hyperbolic fixed point $0$ has no mixed second order resonances in the contracting coordinates, then it satisfies the normal form condition.

Remark 2.13.

The $C^{\infty}$ assumption in the above Lemma 2.12 can be replaced with a $C^{k}$ assumption, where $k$ depends on the spectrum of the linear part, like in Bronstein and Kopanskii [2] (Theorem 11.9). As this dependence is a bit complicated we assume $C^{\infty}$ .

Proof.

Let $x=(x_{1},\dots,x_{p})\in{\mathbb{R}}^{p}$ , $y=(y_{1},\dots,y_{m})\in{\mathbb{R}}^{m}$ ( $p+m=n$ ) and $f(x,y):{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ have the linear part

(({\mathcal{A}}x)_{1},\dots,({\mathcal{A}}x)_{p},({\mathcal{B}}y)_{1},\dots,({\mathcal{B}}y)_{m}),

with $\|{\mathcal{A}}^{-1}\|$ , $\|{\mathcal{B}}\|<{\lambda}<1$ .

First, we will establish conjugation between $f$ and the following normal form:

{\mathcal{A}}x+\left(\sum_{{i=1,\dots,p;}\atop{j=1,\dots,m}}a_{ij}^{1}x_{i}y_{j},\dots,\sum_{{i=1,\dots,p;}\atop{j=1,\dots,m}}a_{ij}^{p}x_{i}y_{j}\right),

{\mathcal{B}}y+\left(\sum_{{i=1,\dots,p;}\atop{j=1,\dots,m}}b_{ij}^{1}x_{i}y_{j},\dots,\sum_{{i=1,\dots,p;}\atop{j=1,\dots,m}}b_{ij}^{m}x_{i}y_{j}\right).

Conjugation of $f$ with this quadratic polynomial can be constructed $C^{1}$ -smooth, because any term of degree higher than 2 in the polynomial expansion, of a $C^{\infty}$ diffeomorphism with a hyperbolic fixed point satisfies the $S(1)$ condition defined by Bronstein and Kopanskii [2] (Definition 7.4, page 110).

Suppose that the polynomial expansion of $f$ contains a term $cx^{k}y^{l}$ and at least one of the two inequalities holds: $|k|>1$ , $|l|>1$ . For definiteness let us assume that $|k|>1$ (where $|k|$ denotes the length of the multiindex $k$ ). Also, we can assume that the eigenvalues are ordered: ${\mathcal{A}}_{1}\leq\dots\leq{\mathcal{A}}_{p}$ . Let $k=(k_{1},\dots,k_{p})$ and $j=\max\{i:k_{i}\neq 0\}$ . Then,

{\mathcal{A}}_{1}^{k_{1}}\cdots{\mathcal{A}}_{j}^{k_{j}}>{\mathcal{A}}_{j},

i.e. $cx^{k}y^{l}$ satisfies the $S(1)$ condition.

Remark 7.6 in Bronstein and Kopanskii [2] (page 111) asserts that $S(1)\implies{\mathcal{A}}(1)$ . Then, Samovol’s theorem ([2], Theorem 10.1, page 179) implies $C^{1}$ -conjugation of $f$ with the quadratic polynomial.

Moreover, since $f$ has no mixed second order resonances in its contracting coordinates, by Samovol’s theorem it can actually be $C^{1}$ -conjugated to

	$\displaystyle{\mathcal{A}}x$	$\displaystyle+\left(\sum_{{i=1,\dots,p;}\atop{j=1,\dots,m}}a_{ij}^{1}x_{i}y_{j},\dots,\sum_{{i=1,\dots,p;}\atop{j=1,\dots,m}}a_{ij}^{p}x_{i}y_{j}\right),$
	$\displaystyle{\mathcal{B}}y.$

∎

Obviously, the conclusion of the $\lambda$ -Lemma of Palis is not true for an arbitrary degenerate (non-transverse) crossing (see, for example, figure 4).

We will now prove an analog of the ${\lambda}$ -Lemma for the non-transverse case in ${\mathbb{R}}^{n}$ .

Lemma 2.14 (Singular $\lambda$ -Lemma).

Let $f$ be a $C^{2}$ diffeomorphism of ${\mathbb{R}}^{n}$ with a hyperbolic fixed point at $0$ and with $m$ - and $p$ -dimensional stable and unstable manifolds $W_{S}$ and $W_{U}$ ( $m+p=n$ ), respectively. Assume that $f$ satisfies the normal form condition and there exists $\Lambda\subset W_{U}$ , a graph portion in a small neighborhood of $0$ , associated with some degenerate homoclinic point $A$ (cf. Definition 2.8). Also assume that $l$ , the order of contact at $A$ , is finite ( $1<l<\infty$ ).

Then, for any $\rho>0$ , for an arbitrarily small $\epsilon$ -neighborhood ${\mathcal{U}}\subset{\mathbb{R}}^{n}$ of the origin and for the graph portion ${\Lambda}$ , the set $(\bigcup_{n\geq 0}f^{n}({\Lambda}))\setminus{\mathcal{U}}$ contains disks $\rho$ - $C^{1}$ close to ${\mathcal{V}}\setminus{\mathcal{U}}$ (here ${\mathcal{V}}$ is a small neighborhood of $0$ in the local unstable manifold).

\psfrag{L}{$\Lambda$}\psfrag{f}{$f(\Lambda)$}\psfrag{ff}{$f^{2}(\Lambda)$}\psfrag{A}{$A$}\psfrag{f(A)}{$f(A)$}\psfrag{f(f(A))}{$f^{2}(A)$}\psfrag{V}{$\mathcal{V}$}\psfrag{S}{$W_{S}$}\psfrag{U}{$W_{U}$}\psfrag{X}{$X$}\psfrag{Y}{$Y$}\psfrag{o}{$0$}\epsfbox{lemma.eps}

Figure 2. Iterations of the graph portion

\Lambda

with the diffeomorphism

f

Proof.

Let $\alpha=1/l$ ( $0<\alpha<1$ ). Since ${\Lambda}$ is a graph portion that has order of contact $l$ with $W_{S}$ at point $A$ , we can assume that locally ${\Lambda}$ is the graph of a function

{\Lambda}(x)=A+r(x):{\mathbb{R}}^{p}\to{\mathbb{R}}^{n},\quad r(0)=0,

and for any sufficiently small $\sigma>0$

|r(x)|\leq\mathop{\rm const}\cdot|x|^{\alpha}\quad\mbox{and}\quad\left|\frac{\partial}{\partial x_{i}}r(x)\right|\leq\mathop{\rm const}\cdot|x|^{\alpha-1}\

for all $|x|<\sigma$ , $i=1,\dots,p$ . Let $x=(x_{1},\dots,x_{p})\in{\mathbb{R}}^{p}$ , $y=(y_{1},\dots,y_{m})\in{\mathbb{R}}^{m}$ ( $p+m=n$ ) and $f(x,y):{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ have the linear part

(({\mathcal{A}}x)_{1},\dots,({\mathcal{A}}x)_{p},({\mathcal{B}}y)_{1},\dots,({\mathcal{B}}y)_{m}).

Assume that $\|{\mathcal{A}}^{-1}\|$ , $\|{\mathcal{B}}\|<{\lambda}<1$ . Choose an arbitrarily small $\Delta$ . We know that locally $f$ can be written in the form $f(x,y)=(S_{1}(x,y),S_{2}(x,y))$ , where $S_{2}(x,y)$ is linear, $S_{2}(x,y)={\mathcal{B}}y$ . Since $f$ is $C^{2}$ , the existence of invariant manifolds implies that, in appropriate coordinates and in a sufficiently small neighborhood of the origin, $f$ can be written as

	$\displaystyle S_{1}(x,y)={\mathcal{A}}x$	$\displaystyle+\left(\sum_{i=1,\dots,p}x_{i}U_{i}^{1}(x,y),\dots,\sum_{i=1,\dots,p}x_{i}U_{i}^{p}(x,y)\right),$
	$\displaystyle S_{2}(x,y)={\mathcal{B}}y,$

with $U(0)=0$ , $\|U\|_{C^{0}}\leq\Delta$ (for an arbitrary small $\Delta$ ), and $\|U\|_{C^{1}}$ bounded.

Consider $f(x,{\Lambda}(x))=(T_{1}^{\Lambda}(x),T_{2}^{\Lambda}(x))$ . We will work with $(x,T_{2}^{\Lambda}\circ(T_{1}^{\Lambda})^{-1}(x))$ and deduce that $f^{n}(x,{\Lambda}(x))$ is $C^{1}$ -small for $n$ big enough, $|x|\in(\epsilon,\sigma)$ , and $\sigma>0$ sufficiently small. First we will show that, in $C^{1}$ -topology, $(T_{1}^{\Lambda})^{-1}$ is $\Delta$ -close to ${\mathcal{A}}^{-1}$ . For simplicity, we will denote $T_{1}^{\Lambda}$ by $T_{1}$ and $T_{2}^{\Lambda}$ by $T_{2}$ . Then

T_{1}(x)={\mathcal{A}}x+\left(\sum_{i=1,\dots,p}x_{i}U_{i}^{1}(x,{\Lambda}(x)),\dots,\sum_{i=1,\dots,p}x_{i}U_{i}^{p}(x,{\Lambda}(x))\right).

Claim 2.15.

\bigl{\|}T_{1}^{-n}\bigr{\|}_{C^{1}}\leq\left(\bigl{\|}A^{-1}\bigr{\|}_{C^{1}}+K\cdot\Delta\right)^{n}

for $|x|<\sigma$ ( $\sigma>0$ sufficiently small, $K>0$ ).

Proof.

Fix some $l\in\{1,\dots,p\}$ . Recall that $\Lambda(x)=A+r(x)$ .

	$\displaystyle\bigg{\|}x_{i}\frac{\partial}{\partial x_{l}}{\Lambda}_{j}(x)\bigg{\|}$	$\displaystyle\leq\|x_{i}\|\cdot\bigg{\|}\frac{\partial}{\partial x_{l}}{\Lambda}_{j}(x)\bigg{\|}$
		$\displaystyle\leq\|x\|\cdot O(1)\|x\|^{\alpha-1}$
		$\displaystyle\leq O(1)\|x\|^{\alpha}$

Through the proof of this Theorem, $O(1)$ will denote the set of functions

	$\displaystyle O(1)=$	$\displaystyle\big{\{}\gamma:{\mathbb{R}}\mapsto{\mathbb{R}}\mbox{, such that there exists a positive constant $c$ with }$
		$\displaystyle\|\gamma(\zeta)\|\leq c\mbox{ for all sufficiently small }\zeta\big{\}}$

Also,

	$\displaystyle\|x\|\cdot\bigg{\|}\frac{\partial}{\partial x_{l}}U_{i}^{t}(x,{\Lambda}(x))\bigg{\|}$	$\displaystyle=\|x\|\cdot\bigg{\|}\frac{\partial}{\partial x_{l}}U_{i}^{t}(x,y)+\sum_{k=1}^{m}\frac{\partial}{\partial y_{k}}U_{i}^{t}(x,y)\cdot\frac{\partial}{\partial x_{l}}{\Lambda}_{k}(x)\bigg{\|}$
		$\displaystyle=O(1)\|x\|^{\alpha},$

and

\big{\|}U(x,{\Lambda}(x))\big{\|}_{C^{0}}\leq\Delta.

Therefore,

\displaystyle\Bigg{\|}\Bigg{(}\sum_{i=1,\dots,p}x_{i}U_{i}^{1}(x,{\Lambda}(x)),\dots,\sum_{i=1,\dots,p}x_{i}U_{i}^{p}(x,{\Lambda}(x))\Bigg{)}\Bigg{\|}_{C^{1}}\leq\Delta\cdot O(1),

if $\sigma$ is sufficiently small and $|x|<\sigma$ . (An arbitrarily small $\Delta$ was chosen earlier). Then,

\bigl{\|}T_{1}^{-n}\bigr{\|}_{C^{1}}\leq\left(\bigl{\|}A^{-1}\bigr{\|}_{C^{1}}+K\cdot\Delta\right)^{n}

∎

Now, we continue the proof of Lemma 2.14 and show that $\|T_{2}^{n}\circ T_{1}^{-n}\|_{C^{1}}$ is $\rho$ -small.

	$\displaystyle\\|T_{2}^{n}\circ T_{1}^{-n}\\|_{C^{1}}$	$\displaystyle=\\|{\mathcal{B}}^{n}\cdot\Lambda(T_{1}^{-n})\\|_{C^{1}}$
		$\displaystyle\leq\\|{\mathcal{B}}\\|^{n}\cdot\\|\Lambda(T_{1}^{-n})\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\\|(T_{1}^{-n})^{\alpha-1}\\|_{C^{0}}\cdot\\|T_{1}^{-n}\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\\|T_{1}^{-n}\\|_{C^{0}}^{\alpha-1}\cdot\\|T_{1}^{-n}\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\big{(}\\|T_{1}^{-n}\\|_{C^{1}}\\|x\\|\big{)}^{\alpha-1}\cdot\\|T_{1}^{-n}\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\\|T_{1}^{-n}\\|_{C^{1}}^{\alpha}\cdot\\|x\\|^{\alpha-1}.$

Applying Claim 2.15 to the last expression, we obtain

\|T_{2}^{n}\circ T_{1}^{-n}\|_{C^{1}}=O(1)\cdot\Big{(}\big{\|}A^{-1}\big{\|}+\Delta K\Big{)}^{\alpha n}\big{\|}x\big{\|}^{\alpha-1}.

The last expression is small if $\big{\|}x\big{\|}>\epsilon$ and $n$ is big enough. ∎

\psfrag{q}{$A$}\psfrag{f(f(q))}{$f^{2}(A)$}\psfrag{f(q)}{$f(A)$}\psfrag{p}{$0$}\epsfbox{lemma_counter_ex.eps}

Figure 3. The iterated manifold is not a graph portion. The iterations of this manifold do not come

C^{1}

-close to the neighborhood of 0 in

W_{U}

Remark 2.16.

Clearly, if $W_{U}$ does not contain any graph portion $\Lambda$ , the conclusion of the $\lambda$ -Lemma 2.14 is not true. Figure 3 illustrates this situation.

3. The Main Theorem

In this section, we extend the results of Hirsch [9], Homburg and Weiss [10], and Gavrilov and Shilnikov [5], [6] to certain types of non-planar homoclinic tangencies with finite orders of contact. Our goal is to establish a transversal crossing arbitrarily close to the non-transversal one. Then, it would follow that there exist horseshoes, located arbitrarily close to the degenerate crossing.

Consider a diffeomorphism $f$ on ${\mathbb{R}}^{n}$ which has $1$ -dimensional stable and $(n-1)$ -dimensional unstable manifolds, and a homoclinic point $B$ . Since the stable manifold is $1$ -dimensional, we can obviously define one- and two-sided homoclinic intersections.

Definition 3.1.

We will say that $f$ has a homoclinic tangency of Newhouse type, if $f$ possesses only one-sided homoclinic tangencies.

If $f$ possesses only one-sided homoclinic tangencies, then transversal intersection need not exist in the neighborhood of $B$ even in the planar case. The Newhouse example (Figure 4) illustrates this situation. In the planar case, this type of tangency (Newhouse type) was studied by Homburg and Weiss [10]. Unfortunately, we can not extend their results to ${\mathbb{R}}^{n}$ .

\psfrag{p}{$p$}\psfrag{q}{$B$}\epsfbox{newhouse.eps}

Figure 4. Newhouse example.

The other three cases of one-sided intersections in ${\mathbb{R}}^{2}$ were studied by Gavrilov and Shilnikov, and Homburg and Weiss (Figure 5, cases I, II, III). The case of two-sided intersection in ${\mathbb{R}}^{2}$ ((Figure 5, case IV) was investigated by Hirsch and Chirchill and Rod.

We will consider, in higher dimensions, the situation similar to the planar cases I, II, III and IV, although we clearly have a different number of cases in ${\mathbb{R}}^{n}$ . For example, cases I and II (with 1-dimensional stable, and 2-dimensional unstable manifolds) are the same in ${\mathbb{R}}^{3}$ . But this is not important for our proof.

If a one-sided intersection is preceded or followed by a two-sided intersection (see Figure 5, cases I, II, III), then we can assume that a two-sided intersection is arbitrarily close to a one-sided touching. Consequently, it is enough to find a transversal crossing arbitrarily close to a two-sided crossing.

\psfrag{1}{case I}\psfrag{2}{case II}\psfrag{3}{case III}\psfrag{4}{case IV}\psfrag{B}{$B$}\epsfbox{twoSided.eps}

Figure 5. Two-sided intersections of invariant manifolds (non-Newhouse types).

Theorem 3.2.

Let $M$ be a smooth manifold, and let $f:M\to M$ be a diffeomorphism with a hyperbolic fixed point $p\in M$ and with stable and unstable invariant manifolds $W_{S}$ and $W_{U}$ at $p$ . Assume:

(i)

$f$ is locally $C^{1}$ -linearizable in a neighborhood of $p$ ;
(ii)

$W_{U}$ and $W_{S}$ have complementary dimensions $n-1$ and $1$ ;
(iii)

$W_{U}$ and $W_{S}$ have a finite tangential contact at an isolated homoclinic point $B\in M$ ( $B\neq p$ ), which is not of Newhouse type;
(iv)

$W_{U}$ contains a graph portion $\Lambda$ associated with $B$ in a small neighborhood, contained in the neighborhood of linearization.

Then the invariant manifolds have a point of transverse intersection arbitrarily close to the point $B$ . Therefore, $f$ has a horseshoe (arbitrarily close to $B$ ) according to the Birkhoff-Smale Theorem.

Proof.

Consider a $C^{1}$ linearization $\phi:N^{\prime}\mapsto N$ from a neighborhood $N^{\prime}\subset M$ of $p$ onto a neighborhood $N\subset{\mathbb{R}}^{n}$ of the origin. We choose a linearization such that $f$ has a normal Jordan form in the new coordinates. We may assume that $\phi$ takes the local stable manifold at $p$ to a neighborhood in the $Y$ -axis, and the local unstable manifold to a neighborhood in the $X$ -hyperplane.

There are images of $\phi(B)$ in $N$ , under forward and backward iterates of $f$ , on both the local stable and local unstable manifolds of $0$ . There is a compact curve, $\Gamma\subset N$ (see Figure 6), with the following properties:

•

$\Gamma$ is an image under $f$ of the stable manifold;
•

$\Gamma$ has finite contact with the $X$ -plane at some isolated point
$A=(A_{1},...,A_{n-1},0)$ ;
•

$\Gamma$ has two-sided intersection with the $X$ -plane at $A$ ;
•

$A$ is arbitrarily close to the origin.

Any $C^{1}$ -curve $\Gamma$ immersed in ${\mathbb{R}}^{n}$ , which has finite order of contact with the $X$ -hyperplane at point $A=(A_{1},...,A_{n-1},0)$ , locally can be parameterized as:

\Gamma=\left\{\begin{array}[]{lll}u^{1}&=&a_{1}(\tau)+A_{1}\\ .&.&.\\ .&.&.\\ .&.&.\\ u^{n-1}&=&a_{n-1}(\tau)+A_{n-1}\\ v&=&{\tau}^{l},\end{array}\right.

for some constants $l>1$ , $A_{j}$ , and $C^{1}$ functions $a_{j}$ , such that $a_{j}(0)=0$ and $a_{j}^{\prime}(\tau)\leq r$ in a small neighborhood of the origin, $\max_{j}|A_{j}|>0$ ( $j=1,...,n-1$ ).

Since $\Lambda$ is a graph portion, it can be parameterized as:

\Lambda=\left\{\begin{array}[]{lll}x^{1}&=&t_{1}\\ .&.&.\\ .&.&.\\ .&.&.\\ x^{n-1}&=&t_{n-1}\\ y&=&\phi(t_{1},...,t_{n-1})+B,\end{array}\right.

for some constant $B\neq 0$ and $C^{1}$ function $\phi$ , $\phi(0)=0$ .

Now it suffices to refer to the following lemma, in order to have the proof of this theorem completed.

Lemma 3.3.

Suppose

f:(x,y)\to(\beta_{1}x^{1}+\sigma_{2}x^{2},\beta_{2}x^{2}+\sigma_{3}x^{3},...,\beta_{n-2}x^{n-2}+\sigma_{n-1}x^{n-1},\beta_{n-1}x^{n-1},\alpha y),

0<\alpha<1<\beta_{j}\quad(j=1,...,n-1)

and $\sigma_{k}$ are $0$ or $1$ – according to the Jordan normal form of $f$ .

Then there exists a natural number $k_{*}$ such that $f^{k}\Lambda$ crosses $\Gamma$ transversely if $k>k_{*}.$

\psfrag{xyn}{$(x_{n},y_{n})$}\psfrag{xy1}{$(x_{1},y_{1})$}\psfrag{f(xyn)}{$f^{n}(x_{n},y_{n})$}\psfrag{f(xy1)}{$f^{n}(x_{1},y_{1})$}\psfrag{T}{$f$}\psfrag{f(l)}{$f(\Lambda)$}\psfrag{f(f(l))}{$f^{2}(\Lambda)$}\psfrag{fn(l)}{$f^{n}(\Lambda)$}\psfrag{G}{$\Gamma$}\psfrag{o}{$0$}\psfrag{A}{$B$}\psfrag{aA}{$\alpha B$}\psfrag{a2A}{$\alpha^{2}B$}\psfrag{anA}{$\alpha^{n}B$}\psfrag{B}{$A$}\psfrag{l}{$\Lambda$}\psfrag{X}{$X$}\psfrag{Y}{$Y$}\epsfbox{par.eps}

Figure 6. Degenerate homoclinic crossing near the fixed point

0

. Illustration of the proof.

Proof.

To simplify notations, first assume that $f$ has a diagonal Jordan normal form, i.e., all $\sigma_{k}$ are 0. Consider the iterations of $\Lambda$ by $f$ :

f^{k_{i}}\Lambda=\left\{\begin{array}[]{lll}u^{1}&=&\beta_{1}^{k_{i}}t_{1}\\ .&.&.\\ .&.&.\\ .&.&.\\ u^{n-1}&=&\beta_{n-1}^{k_{i}}t_{n-1}\\ v&=&\alpha^{k_{i}}\left(\phi(t_{1},...,t_{n-1})+B\right)\\ \end{array}\right.

We can find sequences $(x_{i},y_{i})\in\Lambda$ and $k_{i}\rightarrow\infty$ such that $(x_{i},y_{i})\rightarrow(0,B)$ and $f^{k_{i}}(x_{i},y_{i})\rightarrow(A,0)$ . This implies that $f^{k_{i}}\Lambda$ and $\Gamma$ will cross at some point other than $(0,B)$ and $(A,0)$ (see Figure 6). Thus, we only need to show that this intersection is transversal.

Consider the matrix, constructed in the following way:

•

The first $n-1$ rows represent the linear space of the tangent hyperplane to the $(n-1)$ -manifold $f^{k_{i}}\Lambda$ .
•

The last row represents the linear space of the tangent line to the curve $\Gamma$ .

In order to prove that $f^{k_{i}}\Lambda$ crosses $\Gamma$ transversely at some points of the form $(u^{1}(t_{i}),...,u^{n-1}(t_{i}),v(t_{i}))$ , we show that the determinant of the matrix below is never $0$ for ${k_{i}}$ sufficiently large. We consider the following determinant $D_{i}$ , which corresponds to the $k_{i}$ -th iteration of $f$ :

D_{i}=\left|\begin{array}[]{ccccc}\beta_{1}^{k_{i}}&&0&&\alpha^{k_{i}}\frac{\partial}{\partial t_{1}}\phi(t_{i})\\ &&&&\\ &&&&\\ &\ddots&&&\vdots\\ &&&&\\ 0&&\beta_{n-1}^{k_{i}}&&\alpha^{k_{i}}\frac{\partial}{\partial t_{n-1}}\phi(t_{i})\\ &&&&\\ &&&&\\ a_{1}^{\prime}(\tau_{i})&\ldots&a_{n-1}^{\prime}(\tau_{i})&&l\tau_{i}^{l-1}\\ \end{array}\right|

Since $\Lambda$ intersects $\Gamma$ ,

a_{j}(\tau_{i})+A_{j}=\beta_{j}^{k_{i}}t_{i},\ \ \ \ \ \ \ \ j=1,...,n-1

and

\tau_{i}^{l}=\alpha^{k_{i}}(\phi(t_{i})+B).

Then, it follows that

\tau^{l-1}=\alpha^{k_{i}\frac{l-1}{l}}\left(\phi(t(\tau))+B\right)^{\frac{l-1}{l}}

for any small value of the parameter $\tau$ , and the corresponding small value of the parameter $t(\tau)$ .

We substitute the above expression in the matrix $D_{i}$ and calculate its determinant:

D_{i}=\beta_{1}^{k_{i}}...\beta_{n-1}^{k_{i}}\Bigg{[}(-1)^{n-1}l\alpha^{k_{i}\frac{l-1}{l}}(\phi(t(\tau))+B)^{\frac{l-1}{l}}

\left.+\sum_{j=1}^{n-1}\alpha^{k_{i}}\frac{\partial}{\partial t_{j}}\phi(t(\tau))(-1)^{j}a_{j}^{\prime}(\tau)\beta_{j}^{-k_{i}}\right]

We can assume that $\phi(t(\tau))+B>B/2$ . Also, $a_{j}^{\prime}(\tau)$ in our parameterization of $\Gamma$ are bounded from above by $r$ .

By Lemma 2.14 for any $\epsilon>0$ there exists $k_{*}$ such that for any $k>k_{*}$

\alpha^{k}\frac{\partial}{\partial t_{j}}\phi(t(\tau))<\epsilon,

for all $j=1,...n-1$ . Then

|D_{i}|>\beta_{1}^{k_{i}}...\beta_{n-1}^{k_{i}}\left[l\alpha^{k_{i}\frac{l-1}{l}}(B/2)^{\frac{l-1}{l}}-\alpha^{k_{i}}\epsilon(n-1)r\beta_{j}^{-k_{i}}\right]

=\beta_{1}^{k_{i}}...\beta_{n-1}^{k_{i}}\alpha^{k_{i}\frac{l-1}{l}}\left[l(B/2)^{\frac{l-1}{l}}-\alpha^{k_{i}/l}\epsilon(n-1)r\beta_{j}^{-k_{i}}\right]

Take

\epsilon<\frac{l(B/2)^{\frac{l-1}{l}}}{r(n-1)}

Then,

\left[l(B/2)^{\frac{l-1}{l}}-\alpha^{k_{i}/l}\epsilon(n-1)r\beta_{j}^{-k_{i}}\right]

is positive for any sufficiently big iteration $k_{i}$ .

If $f$ is non-diagonalizable, using the Jordan normal form, one can see that

D_{i}=\beta_{1}^{k_{i}}...\beta_{n-1}^{k_{i}}\left[(-1)^{n-1}l\alpha^{k_{i}\frac{l-1}{l}}(\phi(t(\tau))+B)^{\frac{l-1}{l}}\right.

\left.+\sum_{j=1}^{n-1}\alpha^{k_{i}}\frac{\partial}{\partial t_{j}}\phi(t(\tau))(-1)^{j}a_{j}^{\prime}(\tau)\beta_{j}^{-k_{i}}\right.

\left.-\sum_{j=1}^{n-1}\alpha^{k_{i}}\frac{\partial}{\partial t_{j}}\phi(t(\tau))P_{k_{i}-1}(\beta_{j})\beta_{j}^{-k_{i}}\right],

where $P_{k_{i}-1}(\beta_{j})$ are polynomial functions of degree $k_{i}-1$ .
Then,

|D_{i}|>\beta_{1}^{k_{i}}...\beta_{n-1}^{k_{i}}\alpha^{k_{i}\frac{l-1}{l}}\left[l(B/2)^{\frac{l-1}{l}}-\right.

\left.\left(\alpha^{k_{i}/l}\epsilon(n-1)r\beta_{j}^{-k_{i}}+\alpha^{k_{i}/l}\epsilon(n-1)\left|P_{k_{i}-1}(\beta_{j})\right|\beta_{j}^{-k_{i}}\right)\right]

and

\left[l(B/2)^{\frac{l-1}{l}}-\left(\alpha^{k_{i}/l}\epsilon(n-1)r\beta_{j}^{-k_{i}}+\alpha^{k_{i}/l}\epsilon(n-1)\left|P_{k_{i}-1}(\beta_{j})\right|\beta_{j}^{-k_{i}}\right)\right]

is also non-zero, because $\alpha^{k_{i}/l}$ tends to $0$ exponentially, as $k_{i}\to\infty$ .

This implies that for some $k_{*}$ , the iterations $f^{k_{i}}$ ( $k_{i}>k_{*}$ ) applied to the manifold $\Lambda$ near the origin will produce a transverse crossing with $\Gamma$ .

∎

Applying Lemma 3.3, we complete the proof of Theorem 3.2. ∎

\psfrag{p}{$0$}\psfrag{B}{$A$}\psfrag{U}{$W_{U}$}\psfrag{S}{$W_{S}$}\psfrag{L}{$\Lambda$}\psfrag{fL}{$f(\Lambda)$}\psfrag{ffL}{$f^{2}(\Lambda)$}\psfrag{G}{$\Gamma$}\epsfbox{theorem_counter_ex.eps}

Figure 7.

\Lambda

is not a graph portion. Intersections of

\Gamma

with

f^{n}(\Lambda)

are indicated by a thicker line.

\Gamma

and

f^{n}(\Lambda)

have no transversal intersections.

Remark 3.4.

It is clear from the proof of Theorem 3.2 that, if $W_{U}$ does not contain any graph portion, then $W_{U}$ possibly has no transversal crossings with $\Gamma$ . Figure 7 shows that $\Gamma\cap f^{n}(\Lambda)$ can be a 1-dimensional manifold.

References

[1] V. I. Arnold, S. M. Guseĭn-Zade, A. N. Varchenko, Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts, Birkhäuser, Boston, 1985.
[2] I. U. Bronstein, A. Ya. Kopanskii, Smooth Invariant Manifolds and Normal Forms, World Scientific, Singapore, 1994.
[3] K. Burns, H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys., 172 (1995), 95-118.
[4] R. Churchill and D. Rod, Pathology in dynamical systems. III. Analytic Hamiltonians, J. Diff. Equations, 37 (1980), 23-38.
[5] N. Gavrilov, L. Silnikov On 3-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. I, Math. USSR Sb., 88 (1972), 467-485.
[6] N. Gavrilov, L. Silnikov On 3-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve, Math. USSR Sb., 90 (1973), 139-156.
[7] S. Gonchenko, L. Shilnikov Dynamical systems with structurally unstable homoclinic curves, Soviet Math. Dokl., 33 (1986), 234-238.
[8] P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2), 5 (1960), 220-241.
[9] M. Hirsch, Degenerate homoclinic crossings in surface diffeomorphisms, preprint (1993).
[10] A. J. Homburg, H. Weiss, A geometric criterion for positive topological entropy. II: Homoclinic tangencies, Comm. Math. Phys., 208 (1999), 267-273.
[11] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES, 51 (1980), 137-173.
[12] A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, Proc. International Congress of Mathematicians, Warszawa, 2 (1983), 1245-1254.
[13] J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math., 9 (1956), 673-692.
[14] S. Newhouse and J. Palis, Cycles and bifurcation theory, Asterisque, 31, Soc. Math. France, Paris, 43-140 (1976).
[15] J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-405.
[16] V. Rayskin, Multidimensional singular $\lambda$ -Lemma, Electron. J. Diff. Eqns., 38 (2003), 1-9.
[17] S. Smale, Diffeomorphisms with many periodic points, in:Differential and Combinatorial topology (edited by S. S. Cairnes), Princeton University Press, 63-80, (1965).
[18] R. Uribe-Vargas, Four-vertex theorems in higher dimensional spaces for a larger class of curves than the convex ones, C.R. Acad. Sci. Paris, Ser. I, 330 (2000), 1085-1090.

Received October 2003; revised September 2004.

	$\displaystyle\bigg{\|}x_{i}\frac{\partial}{\partial x_{l}}{\Lambda}_{j}(x)\bigg{\|}$	$\displaystyle\leq\|x_{i}\|\cdot\bigg{\|}\frac{\partial}{\partial x_{l}}{\Lambda}_{j}(x)\bigg{\|}$
		$\displaystyle\leq\|x\|\cdot O(1)\|x\|^{\alpha-1}$
		$\displaystyle\leq O(1)\|x\|^{\alpha}$

	$\displaystyle\\|T_{2}^{n}\circ T_{1}^{-n}\\|_{C^{1}}$	$\displaystyle=\\|{\mathcal{B}}^{n}\cdot\Lambda(T_{1}^{-n})\\|_{C^{1}}$
		$\displaystyle\leq\\|{\mathcal{B}}\\|^{n}\cdot\\|\Lambda(T_{1}^{-n})\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\\|(T_{1}^{-n})^{\alpha-1}\\|_{C^{0}}\cdot\\|T_{1}^{-n}\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\\|T_{1}^{-n}\\|_{C^{0}}^{\alpha-1}\cdot\\|T_{1}^{-n}\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\big{(}\\|T_{1}^{-n}\\|_{C^{1}}\\|x\\|\big{)}^{\alpha-1}\cdot\\|T_{1}^{-n}\\|_{C^{1}}$
		$\displaystyle=O(1)\cdot\\|T_{1}^{-n}\\|_{C^{1}}^{\alpha}\cdot\\|x\\|^{\alpha-1}.$

Homoclinic tangencies in ℝn{\mathbb{R}}^{n}

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

Theorem 1.1 (Birkhoff-Smale).

2. Definitions and Preliminaries

Definition 2.1.

Proposition 2.2.

Proof.

Remark 2.3.

Remark 2.4.

Lemma 2.5.

Proof.

Definition 2.6.

Theorem 2.7 (λ\lambda-Lemma, Palis).

Definition 2.8.

Definition 2.9.

Definition 2.10.

Definition 2.11.

Lemma 2.12.

Remark 2.13.

Proof.

Lemma 2.14 (Singular λ\lambda-Lemma).

Proof.

Claim 2.15.

Proof.

Remark 2.16.

3. The Main Theorem

Definition 3.1.

Theorem 3.2.

Proof.

Lemma 3.3.

Proof.

Remark 3.4.

References

Homoclinic tangencies in ${\mathbb{R}}^{n}$

Theorem 2.7 ( $\lambda$ -Lemma, Palis).

Lemma 2.14 (Singular $\lambda$ -Lemma).