On invariant measures of "satellite" infinitely renormalizable quadratic polynomials

We consider the dynamics $f:\mathbb{C}\to\mathbb{C}$ of a quadratic polynomial. Up to a linear change of coordinates, $f$ has the form $f_{c}(z)=z^{2}+c$ for some $c\in\mathbb{C}$. In this paper, which is the sequel of [9], we assume that $f$ is infinitely-renormalizable. Moreover, in the main results we assume that $f$ has infinitely many "satellite renormalizations", see e.g. [19], or below for definitions. Dynamics, geometry and topology of such system can be very non-trivial, in particular, due to the fact that different renormalization levels are largely independent.

Historically, the first example of infinitely-renormalizsable one-dimensional map was, probably, the Feigenbaum period-doubling quadratic polynomial $f_{c_{F}}$, where $c_{F}=-1.4...$ [6]. The Julia set of $f_{c_{F}}$ is locally connected [7] as it follows from so-called "complex bounds", a compactness property of renormalizations. This is a key tool since [28], in particular, in proving the Feigenbaum-Coullet-Tresser universality conjecture [28, 20, 15]. Perhaps, more striking for us are Douady-Hubbard’s examples, or alike, of infinitely-renormalizable quadratic polynomials with non-locally connected Julia sets [17, 27, 10, 11, 12, 4, 3]. As for the Feigenbaum polynomial $f_{c_{F}}$, all the renormalizations of such maps are satellite, although, contrary to $f_{c_{F}}$, combinatorics is unbounded (which, in turn, implies that those maps cannot have complex bounds [1]).

Dynamics of every holomorphic endomorphism of the Riemann sphere $g:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ classically splits $\hat{\mathbb{C}}$ into two subsets: the Fatou set $F(g)$ and its complement the Julia set $J(g)$, where $F(g)$ is the maximal (possibly, empty) open set where the sequence of iterates $g^{n}$, $n=0,1,...$ forms a normal (i.e., a precompact) family. See e.g. [2], [16] for the Fatou-Julia theory and [26] for a recent survey.

If $g$ is a polynomial, then the Julia set $J(g)$ coincides with the boundary of the basin of infinity $A(\infty)=\{z\in\mathbb{C}|\lim_{n\to\infty}g^{n}(z)=\infty\}$ of $g$. The complement $\mathbb{C}\setminus A(g)$ is called the filled Julia set $K(g)$ of the polynomial $g$. The compact $K(g)\subset\mathbb{C}$ is connected if and only if it contains all critical points of $g$ in the complex plane.

A quadratic polynomial $f_{c}$ with connected filled Julia set $K(f)$ is renormalizable if, for some topological disks $U\Subset V$ around the critical point $0$ of $f_{c}$, and some $p\geq 2$ (period of the renormalization), the restriction $F:=f_{c}^{p}:U\to V$ is a proper branched covering map (called polynomial-like map) of degree $2$ and the non-escaping set $K(F)=\{z\in U:F^{n}(z)\in U\mbox{ for all }n\geq 1\}$ (called the filled Julia set of the polynomial-like map $F$) is connected. The map $F:U\to V$ is then a renormalization of $f_{c}$ and the set $K(F)$ is a "small" (filled) Julia set of $f_{c}$. By the theory of polynomial-like mappings [5], there is a quasiconformal homeomorphism of $\mathbb{C}$, which is conformal on $K(F)$, that conjugates $F$ on a neighborhood of $K(F)$ to a uniquely defined another quadratic polynomial $f_{c^{\prime}}$ with connected filled Julia set. If $f_{c^{\prime}}$ is renormalizable by itself, then $f_{c}$ is called twice renormalizable, etc. If $f_{c}$ admits infinitely many renormalizations, it is called infinitely-renormalizable. The renormalization $F=f^{p}_{c}$ is simple if any two sets $f^{i}(K(f))$, $f^{j}(K(F))$, $0\leq i<j\leq p-1$, are either disjoint or intersect each other at a unique point which does not separate either of them. A simple renormalization $f^{p_{n}}$ is called primitive if all sets $f^{i}(K_{n})$, $i=0,\cdots,p_{n}-1$, are disjoints and satellite otherwise.

To state our main results, Theorems 1.1, let $f(z)=z^{2}+c$ be infinitely renormalizable. Then its Julia set $J=J(f)$ coincides with the filled Julia set $K(f)$ and is a nowhere dense compact full connected subset of $\mathbb{C}$. Let $1=p_{0}<p_{1}<...<p_{n}<...$ be the sequence of consecutive periods of simple renormalizations of $f$ and $J_{n}\ni 0$ denote the "small" Julia set of the $n$-renormalization (where $J_{0}=J$). Then $p_{n+1}/p_{n}$ is an integer, $f^{p_{n}}(J_{n})=J_{n}$, for any $n$, and $f$-orbits of $J_{n}$,

$n=0,1,...$, form a strictly decreasing sequence of compact subsets of $\mathbb{C}$. Let

be the intersection of the orbits of the "small" Julia sets $J_{n}$. For every $n$, repelling periodic orbits of $f$ are dense in $orb(J_{n})$, while each component of $J_{\infty}$ is wandering. In particular, $J_{\infty}$ contains no periodic points of $f$.

Let

be the postcritical set of $f$. Clearly,

Moreover, the critical point $0$ is recurrent, hence,

where $\omega(z)$ is the omega-limit set of a point $z\in J$.

We prove in [9] that $J_{\infty}$ cannot contain any hyperbolic set. On the other hand, a hyperbolic set of a rational map always carries an invariant measure with a positive Lyapunov exponent. So a generalization of [9] would be that $J_{\infty}$ never carries such a measure. Here we prove this generalization for a class of "satellite" infinitely-renormalizable quadratic polynomials:

Let us comment on the behavior of the restriction map $f:J_{\infty}\to J_{\infty}$ where $f$ as in Theorem 1.1. First, by [9], the postcritical set $P$ must intersect the omega-limit set $\omega(x)$ of each $x\in J_{\infty}$. At the same time, dynamics and topology of the further restriction $f:P\to P$ can vary. Indeed, there are infinitely renormalizable quadratic polynomials $f$ with all renormalizations being of satellite type such that at least one of the following holds¹¹1A more complete description of $f:P\to P$ should follow from the methods developed in [3].:

(1) $f:P\to P$ is not minimal. This case happens in Douady-Hubbard’s type examples. Indeed, by the basic construction [17], $J_{\infty}$ then contains a closed invariant set $X$ (which is the limit set for the collection of $\alpha$-fixed points of renormalizations) such that $0\notin X$. By [9], $X\cap P$ is non-empty. Thus $X\cap P$ is an invariant non-empty proper compact subset of $P$.

(2) $P$ is a so-called "hairy" Cantor set, in particular, $P$ contains uncountably many non-trivial continua. This case takes place following [3].

(3) $P$ is a Cantor set and $f:P\to P$ is minimal; this happens whenever $f$ either admits complex bounds (which then imply $J_{\infty}=P$) or is robust [19]²²2The ”robustness” can happen without ”complex bounds” as it follows from [3] combined with [1].. Under either of the two conditions, $f:P\to P$ is a minimal homeomorphism, which is topologically conjugate to $x\mapsto x+1$ acting on the projective limit of the sequence of groups $\{\mathbb{Z}/p_{n}\mathbb{Z}\}_{n=1}^{\infty}$; in particular, $f:P\to P$ (hence, also $f:J_{\infty}\to J_{\infty}$, as it follows from the next Corollary 1.3) is uniquely ergodic in this case.

Theorem 1.1 yields the following dichotomy about the measurable dynamics of $f:J\to J$ on the Julia set $J$ of $f$. Recall that, by [22], any invariant probability measure on the Julia set of a rational function has non-negative exponents.

In particular, the set $J_{\infty}\setminus P$ is "measure invisible", see also Proposition 6.1 which is a somewhat stronger version of Corollary 1.3.

For the proof of Corollaries 1.3-1.4, see Sect. 6. The proof of Theorem 1.1 occupies sections 2-5.

As in [9], we use heavily a general result of [23] on the accessibility although the main idea of the proof is different. Indeed, in [9] we utilize the fact that the map cannot be one-to-one on an infinite hyperbolic set. At the present paper, to prove Theorem 1.1 we assign, loosely speaking, an external ray to a typical point of a hypothetical measure with positive exponent such that the family of such rays is invariant and has a controlled geometry. Given a satellite renormalization $f^{p_{n}}$ we use the measure and the above family of rays to choose a point $x$ and build a special domain that covers a "small" Julia set $J_{n,x}\ni x$ such that there is a univalent pullback of the domain by $f^{p_{n}}$ along the renormalization that enters into itself, leading to a contradiction. The choice of $x$ is ’probabilistic’, i.e., made from sets of positive measure, and the construction of the domain differs substantially depending on whether all satellite renormalizations of $f$ are doubling or not.

Acknowledgment. The conclusion (1.2) of Corollary 1.4 that the Lyapunov exponent at the critical value equals zero answers partly a question by Weixiao Shen³³3 Shen asked the following question, in relation with Corollary 5.5 of [14]: Is it possible that the upper Lyapunov exponent at a critical value $v$ of a polynomial $g$ is positive assuming that $g$ is infinitely-renormalizable around $v$? See also Remark 1.1 which inspired the present work as well as the prior one [9]. The authors thank the referee for careful reading the paper and many helpful comments.

Here we collect, for further references and use throughout the paper, necessary notations and general facts. (A)-(D) are slightly adapted versions of (A)-(D) in Sect. 2, [9] which are either well-known [19], [18], or are proved here.

Let $f(z)=z^{2}+c$ be infinitely renormalizable. We keep the notations of the Introduction.

(A). Let $G$ be the Green function of the basin of infinity $A(\infty)=\{z|f^{n}(z)\to\infty,n\to\infty\}$ of $f$ with the standard normalization at infinity $G(z)=\ln|z|+O(1/|z|)$. The external ray $R_{t}$ of argument $t\in{\bf S^{1}}=\mathbb{R}/\mathbb{Z}$ is a gradient line to the level sets of $G$ that has the (asymptotic) argument $t$ at $\infty$. $G(z)$ is called the (Green) level of $z\in A(\infty)$ and the unique $t$ such that $z\in R_{t}$ is called the (external) argument (or angle) of $z$. A point $z\in J(f)$ is accessible if there is an external ray $R_{t}$ which lands at (i.e., converges to) $z$. Then $t$ is called an (external) argument (angle) of $z$.

Let $\sigma:{\bf S^{1}}\to{\bf S^{1}}$ be the doubling map $\sigma(t)=2t(\mod 1)$. Then $f(R_{t})=R_{\sigma(t)}$.

Every point $a$ of a repelling cycle $O_{a}$ of period $p$ is the landing point of an equal number $v$, $1\leq v<\infty$, of external rays where $v$ coincides with the number of connected components of $J(f)\setminus\{a\}$. Their arguments are permuted by $\sigma^{p}$ according to a rational rotation number $r/q$ (written in the lowest term); $v/q$ is the number of cycles of rays landing at $a$. If $v\geq 2$, there is an alternative [18]:

$r/q=0/1$, then $v=2$ so that each of two external ray landing at $a$ is fixed by $f^{p}$,

$r/q\neq 0/1$, i.e., $q\geq 2$, then $v=q$, i.e., the arguments of $q$ rays landing at $a$ form a single cycle of $\sigma^{p}$.

(B). All periodic points of $f$ are repelling. Given a small Julia set $J_{n}$ containing $0$, sets $f^{j}(J_{n})$, $0\leq j<p_{n}$, are called small Julia sets of level $n$. Each $f^{j}(J_{n})$ contains $p_{n+1}/p_{n}\geq 2$ small Julia sets of level $n+1$. We have $J_{n}=-J_{n}$. Since all renormalizations are simple, for $j\neq 0$, the symmetric companion $-f^{j}(J_{n})$ of $f^{j}(J_{n})$ can intersect the orbit $orb(J_{n})=\cup_{j=0}^{p_{n}-1}f^{j}(J_{n})$ of $J_{n}$ only at a single point which is periodic. On the other hand, since only finitely many external rays converge to each periodic point of $f$, the set $J_{\infty}$ contains no periodic points. In particular, each component $K$ of $J_{\infty}$ is wandering, i.e., $f^{i}(K)\cap f^{j}(K)=\emptyset$ for all $0\leq i<j<\infty$. All this implies that $\{x,-x\}\subset J_{\infty}$ if and only if $x\in K_{0}:=\cap_{n=1}^{\infty}J_{n}$.

Given $x\in J_{\infty}$, for every $n$, let $j_{n}(x)$ be the unique $j\in\{0,1,\cdots,p_{n}-1\}$ such that $x\in f^{j_{n}(x)}(J_{n})$. Let $J_{n,x}=f^{j_{n}(x)}(J_{n})$ be a small Julia set of level $n$ containing $x$ and $K_{x}=\cap_{n\geq 0}J_{n,x}$, a component of $J_{\infty}$ containing $x$.

In particular, $K_{0}=\cap_{n\geq 0}J_{n}$ is the component of $J_{\infty}$ containing $0$ and $K_{c}=\cap_{n=1}^{\infty}f(J_{n})$, the component containing $c$.

Note that either $p_{n}-j_{n}(x)\to\infty$ as $n\to\infty$ or $p_{n}-j_{n}(x)=N$ for some $N\geq 0$ and all $n$, that is, $f^{N}(x)\in K_{0}$. This is so since the sequence of the sets $J_{n}$ is non-increasing, hence $J_{n,x}$ non-increasing, hence $p_{n}-j_{n}(x)$ (the time to reach $J_{n}$) non-decreasing.

The map $f:K_{x}\to K_{f(x)}$ is two-to-one if $x=0$ and one-to-one otherwise. Moreover, for every $y\in J_{\infty}$, $f^{-1}(y)\cap J_{\infty}$ consists of two points if $y\in K_{c}$ and consists of a single point otherwise. Denote

We conclude that:

$f:J_{\infty}^{\prime}\to J_{\infty}^{\prime}$ is a homeomorphism. Given $x\in J_{\infty}^{\prime}$ and $m>0$, denote $x_{m}=f^{m}(x)$ and

that is, the only point $f^{-m}(x)\cap J_{\infty}$.

(C). Given $n\geq 0$, the map $f^{p_{n}}:f(J_{n})\to f(J_{n})$ has two fixed points: the separating fixed point $\alpha_{n}$ (that is, $f(J_{n})\setminus\{\alpha_{n}\}$ has at least two components) and the non-separating $\beta_{n}$ (so that $f(J_{n})\setminus\beta_{n}$ has a single component).

For every $n>0$, there are $0<t_{n}<\tilde{t}_{n}<1$ such that two rays $R_{t_{n}}$ and $R_{\tilde{t}_{n}}$ land at the non-separating fixed point $\beta_{n}\in f(J_{n})$ of $f^{p_{n}}$ and the component $\Omega_{n}$ of ${\bf C}\setminus(R_{t_{n}}\cup R_{\tilde{t}_{n}}\cup\beta_{n})$ which does not contain $0$ has two characteristic propertiers [18]:

(i) $\Omega_{n}$ contains $c$ and is disjoint with the forward orbit of $\beta_{n}$,

(ii) for every $1\leq j<p_{n}$, consider arguments (angles) of external rays which land at $f^{j-1}(\beta_{n})$. The angles split ${\bf S^{1}}$ into finitely many arcs. Then the length of any such arc is bigger than the length of the arc

Denote

The rays $R_{t_{n}^{\prime}}$, $R_{\tilde{t}_{n}^{\prime}}$ land at a common point $\beta_{n}^{\prime}\in f^{-p_{n}}(\beta_{n})\cap\Omega_{n}$. Introduce an (unbounded) domain $U_{n}$ with the boundary to be two curves $R_{t_{n}}\cup R_{\tilde{t}_{n}}\cup\beta_{n}$ and $R_{t_{n}^{\prime}}\cup R_{\tilde{t}_{n}^{\prime}}\cup\beta_{n}^{\prime}$. Then $c\in U_{n}$ and $f^{p_{n}}:U_{n}\to\Omega_{n}$ is a two-to-one branched covering. Also,

Let

so that $s_{n,1}\subset S_{n,1}$ and argument of any ray to $f(J_{n})$ lies in $s_{n,1}$.

Let us iterate this construction. Given $1\leq j\leq p_{n}$, let $S_{n,j}$ be one of the two arcs of ${\bf S^{1}}$ with end points

such that arguments of any ray to $f^{j}(J_{n})$ lies in $S_{n,j}$. Let

where $t_{n,j}^{\prime}=\sigma^{j-1}(t_{n}^{\prime}),\tilde{t}_{n,j}^{\prime}=\sigma^{j-1}(\tilde{t}_{n}^{\prime})$. Then

and argument of any ray to $f^{j}(J_{n})$ lies in fact in $s_{n,j}$. Note that

So $\sigma^{j-1}:s_{n,1}\to s_{n,j}$ is a homeomorphism and $s_{n,j}$ has two components (’windows’) $[t_{n,j},t_{n,j}^{\prime}]$ and $[\tilde{t}_{n,j}^{\prime},\tilde{t}_{n,j}]$ of equal length.

Let $U_{n,j}=f^{j-1}(U_{n})$ and $\beta_{n,j}=f^{j-1}(\beta_{n})$. The domain $U_{n,j}$ is bounded by two rays $R_{t_{n,j}}\cup R_{\tilde{t}_{n,j}}$ converging to $\beta_{n,j}$ and completed by $\beta_{n,j}$ along with two rays $R_{t_{n,j}^{\prime}}\cup R_{\tilde{t}_{n,j}^{\prime}}$ completed by their common limit point $f^{j-1}(\beta_{n}^{\prime})$ where $t_{n,j}^{\prime}=\sigma^{j-1}(t_{n}^{\prime}),\tilde{t}_{n,j}^{\prime}=\sigma^{j-1}(\tilde{t}_{n}^{\prime})$.

By (i)-(ii), for a fixed $n$, domains $U_{n,j}$, $1\leq j\leq p_{n}$, are pairwise disjoint.

Let $U_{n,j-p_{n}}$ be a component of $f^{-(p_{n}-j)}(U_{n})$ which is contained in $U_{n,j}$. Then

is a two-to-one branched covering and

Let $s^{1}_{n,j}$ be the set of arguments of rays entering $U_{n,j-p_{n}}$. Then $s^{1}_{n,j}$ consists of $4$ components so that $\sigma^{p_{n}}$ map homeomorphically each of these components onto one of the ’windows’ of $s_{n,j}$.

Furthermore, let

Unlike the map (2.2), the map

is a two-to-one branched covering only assuming $f^{j-1}:\Omega_{n}\to\Omega_{n,j}$ is a homeomorphism, which holds if and only if $\sigma^{j-1}:S_{n,1}\to\sigma^{j-1}(S_{n,1})$ is a homeomorphism. In the latter case,

Primitive vs satellite renormalizations. Let $n\geq 2$ and $r_{n}/q_{n}$ be the rotation number of $\beta_{n}$. The next claim is well-known, we include the proof for reader’s convenience.

We need a more refined estimate provided the renormalization is not doubling. Assume $f^{p_{n}}$ is satellite so that $p_{n-1}=p_{n}/q_{n}$ with $q_{n}\geq 2$ and the rotation number of $\beta_{n}$ is $r_{n}/q_{n}\neq 0/1$.

(D). Given a compact set $Y\subset J(f)$ denote by $(\tilde{Y})_{f}$ (or simply $\tilde{Y}$, if the map is fixed) the set of arguments of the external rays which have their limit sets contained in $Y$. It follows from (C) that $\tilde{K}_{c}=\bigcap_{n=1}^{\infty}\{[t_{n},t_{n}^{\prime}]\cup[\tilde{t}_{n}^{\prime},\tilde{t}_{n}]\}$, i.e., it is either a single-point set or a two-point set.

Since $\tilde{K}_{c}$ contains at most two angles, $K_{c}$ contains at most two different accessible points. More generally, given $x\in J^{\prime}_{\infty}$ let

Then $s_{n+1,j_{n+1}(x)}\subset s_{n,j_{n}(x)}$ so that

is not empty and consists of either one or two components. Since $p_{n}-j_{n}(x)\to\infty$ for $x\in J^{\prime}_{\infty}$ we conclude using (2.1):

$s_{\infty,x}$ consists of either a single point or two different points. In particular, for any component $K$ of $J_{\infty}$ which is not one of $f^{-j}(K_{0})$, $j\geq 0$, there is either one or two rays tending to $K$.

From now on, $\mu$ is an $f$-invariant probability ergodic measures supported in $J_{\infty}$: $\operatorname{supp}\mu\subset J_{\infty}$, and having a positive Lyapunov exponent

(E). We start with the following basic statement. Parts (i)-(ii) are easy consequences of the invariance of $\mu$ and (B) while (iii) is a part of Pesin’s theory as in [25, Theorem 11.2.3] coupled with the structure of $f:J_{\infty}\to J_{\infty}$, see (B). Recall that $J_{\infty}^{\prime}=J_{\infty}\setminus\cup_{j=-\infty}^{\infty}f^{j}(K_{0})$.