The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,b} \colon z \mapsto ae^z+be^{-z}$ for $a,b\in \mathbb{C}^\ast$, consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and \textit{split} at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$. We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$, and conclude that no two of its dynamic rays land together.
We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called ``rays'', each connecting $\infty$ to a well-defined ``landing point'' in $\C$, so that every point in $\C$ is either on a unique ray or the landing point of finitely many rays. The key features of this paper are the following two: (1) this is the first example of a transcendental dynamical system where the Julia set is all of $\C$ and the dynamics is descri…
We study the dynamics of iterated cosine maps $E\colon z \mapsto ae^z+be^{-z},$ with $a,b \in \C\setminus \{0\}$. We show that the points which converge to infinity under iteration are organized in the form of rays and, as in the exponential family, every escaping point is either on one of these rays or the landing point of a unique ray. Thus we get a complete classification of the escaping points of the cosine family, confirming a conjecture of Eremenko in this case. We also get a particularly strong version of the ``dimension paradox'': the set of rays has Hausdorff dimension 1, while the se…
The purpose of this paper is to establish a topological characterization of all the hyperbolic maps in the Sine family $\{λ\sin(z) \:\big{|}\:λ\ne 0\}$ which have super-attracting cycles.
Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition of the postcritical volume. This allows us to generalize to this setting the theory of stability and bifurcation developed by Berteloot, Dupont and the author for endomorphisms of $\mathbb{P}^k$.
Let f be a meromorphic self-map on a compact Kaehler manifold whose topological degree is strictly larger than the other dynamical degrees. We show that repelling periodic points are equidistributed with respect to the equilibrium measure of f. We also describe the exceptional set of points whose backward orbits are not equidistributed.
Let $(φ_t)$, $(ϕ_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disc $\mathbb D\subset \mathbb C$. Let $f:\mathbb D \to \mathbb D$ be a homeomorphism. We prove that, if $f \circ ϕ_t=φ_t \circ f$ for all $t\geq 0$, then $f$ extends to a homeomorphism of $\bar{\mathbb D}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, $f$ extends to a homeomorphism of $\bar{\mathbb D}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disc.
We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some exam…
We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counte…
We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+κ$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a suitable subset of its set of escaping points, to a restriction of this model dynamics. Furthermore, we show that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's "pinched disk model" for the Julia sets of these maps. On the other hand, we show that two exponential maps are gene…
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the inve…
Topological conjugateness of one dimensional unimodal dynamical systems, which are generated by interval [0, 1] into itself maps are studied. We study the smoothness and differentiability of the conjugacy of symmetrical and non-symmetrical tent maps. Also weprove the extremal property of the length of the graph of the conjugacy of symmetrical and non-symmetrical tent maps.
We prove that a zero topological entropy continuous tree map always displays zero topological sequence entropy when it is restricted to its non-wandering and chain recurrent sets. In addition, we show that a similar result is not possible when the phase space is a dendrite even when we consider only the restriction on the set of periodic points.
We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$ interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map …
We show that a strengthened version of the Collet-Eckmann condition for multimodal maps is topologically invariant. In particular, if f is non-uniformly expanding and the critical points are generic with respect to the absolutely continuous invariant measure, then any map g topologically conjugate to f is also non-uniformly expanding.
This paper is devoted to study the topological invariance of several non-uniform hyperbolicity conditions of one-dimensional maps. In contrast with the case of maps with only one critical point, it is known that for maps with several critical points the Collet-Eckmann condition is not in itself invariance under topological conjugacy. We show that the Collet-Eckmann condition together with any of several slow recurrence conditions is invariant under topological conjugacy. This extends and gives a new proof of a result by Luzzatto and Wang that also applies to the complex setting.
We establish the analogue for maps on trees of the result established in two papers by the first author for interval maps, that a continuous self- map for which all but countably many points have at least m preimages (and none have less than two) has topological entropy bounded below by log m.
Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this paper. This provides a solution for a classical problem in bifurcation theory that was set out three decades ago and remained unexpectedly unpublished until now
We show that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a regular space. Through examples we show that in general composition of topologically expansive homeomorphisms need not be topological expansive and also that conjugate of topologically expansive homeomorphism need not be topological expansive. Further, we obtain a characterization of orbit expansivity in terms of topological expansivity and conclude that if there exists a topologically expansive homeomorphism on a compact uniform space then the space must be metrizable. We also s…
A symmetric Lorenz map is obtain by ``flipping'' one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.
We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we investigate whether they have various properties, including topological transitivity, topological mixing, dense periodic points, and specification. To each Markov multi-map, we associate a shift of finite type (SFT), and then our main results relate the properties of the SFT with those of the Markov multi-map. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse li…
We show that every uniquely ergodic minimal Cantor system is topological orbit equivalent to the natural extension of a numeration scale associated to a logistic map.
We consider a $C^1$ neighborhood of the time-one map of a hyperbolic flow and prove that the topological entropy varies continuously for diffeomorphisms in this neighborhood. This shows that the topological entropy varies continuously for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center bundle.
We present a topological method of obtaining the existence of infinite number of symmetric periodic orbits for systems with reversing symmetry. The method is based on covering relations. We apply the method to a four-dimensional reversible map.
We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map $f$ on a topological graph $G$ has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's Möbius Disjointness Conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no $3$-scrambled tuple if and only if the p…
Topological entropy is a measure of complex dynamics. In this regard, multimodal maps play an important role when it comes to study low-dimensional chaotic dynamics or explain some features of higher dimensional complex dynamics with conceptually simple models. In the first part of this paper an analytical formula for the topological entropy of twice differentiable multimodal maps is derived, and some basic properties are studied. This expression involves the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, its proof leads to a numerical algorithm that…
A numerical algorithm to compute the topological entropy of multimodal maps is proposed. This algorithm results from a closed formula containing the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, it simplifies a previous algorithm, also based on min-max symbols, which was originally proposed for twice differentiable multimodal maps. The new algorithm has been benchmarked against the old one with a number of multimodal maps, the results being reported in the paper. In particular, the comparison is favorable to the new algorithm, except in the unimodal…
We give an upper bound for the topological entropy of maps on inverse limit spaces in terms of their set-valued components. In a special case of a diagonal map on the inverse limit space $\underleftarrow{\lim}(I,f)$, where every diagonal component is the same map $g\colon I\to I$ which strongly commutes with $f$ (i.e. $f^{-1}\circ g=g\circ f^{-1}$), we show that the entropy equals $\max\{\textrm{Ent}(f),\textrm{Ent}(g)\}$. As a side product, we develop some techniques for computing topological entropy of set-valued maps.
We study the non-wandering set of contracting Lorenz maps. We show that if such a map $f$ doesn't have any attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a compact set $Λ$ such that $ω_f(x)=Λ$ for a residual set of points $x \in [0,1]$. Furthermore, we classify the possible kinds of attractors that may occur.
Topological structure and entropy of mixing graph maps
Institute of Mathematics, Jagiellonian University（雅盖隆大学数学研究所）；Faculty of Applied Mathematics, AGH University of Science and Technology（AGH科技大学应用数学学院）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）
Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study a family of generalized Bowen-Series boundary maps and prove the following rigidity result: in this family the topological entropy is constant and depends only on the genus of the surface. We give an explicit formula for this entropy and show that the value of the topological entropy also stays constant in the Teichmüller space of the surface. The proofs use conjugation to maps of constant slope.
We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.
We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is…
We show that every dendrite satisfying the condition that no subtree of it contains all free arcs admits a transitive, even exactly Devaney chaotic map with arbitrarily small entropy. This gives a partial answer to a question of Baldwin from 2001.
We show that graph map with zero topological entropy is Li-Yorke chaotic if and only if it has an NS-pair (a pair of non-separable points containing in a same solenoidal $ω$-limit set), and a non-diagonal pair is an NS-pair if and only if it is an IN-pair if and only if it is an IT-pair. This completes characterization of zero topological sequence entropy for graph maps.
For a circle map $f\colon\mathbb{S}\to\mathbb{S}$ with zero topological entropy, we show that a non-diagonal pair $\langle x,y\rangle\in \mathbb{S}\times \mathbb{S}$ is non-separable if and only if it is an IN-pair if and only if it is an IT-pair. We also show that if a circle map is topological null then the maximal pattern entropy of every open cover is of polynomial order.
We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove that in the class of area contracting maps the signature cannot be a monotone sequence. This explains why classical examples of infinitely renormalizable maps due to Bowen, Franks and Young cannot be achieved by smooth dissipative maps showing that there are topological obstruct…
We show that if (X,T) is a topological dynamical system with is deterministic in the sense of Kamiski, Siemaszko and Szymaski then T^{-1} and the product system need not be determinstic in this sense. However if the product system is deterministic then T^n is deterministic for all integers n>0.
A map $f: \ff^n \to \ff^n$ over a field $\ff$ is called affine if it is of the form $f(x)=Ax+b$, where the matrix $A \in \ff^{n\times n}$ is called the linear part of affine map and $b \in \ff^n$. The affine maps over $\ff=\rr$ or $\cc$ are investigated. We prove that affine maps having fixed points are topologically conjugate if and only if their linear parts are topologically conjugate. If affine maps have no fixed points and $n=1$ or 2, then they are topologically conjugate if and only if their linear parts are either both singular or both non-singular. Thus we obtain classification up to t…
Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $φ: X \mapsto \IR$ a continuous function. We consider the set of points for which the Birkhoff average of $φ$ does not exist (which we call the irregular set for $φ$) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel). We formulate various equivalent natural conditions on $φ$ that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously k…
We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity pieces have disjoint images. The continuity pieces act as stable topological manifolds while the points in the discontinuity lines, separating different continuity pieces, act as topological saddles with an infinite expanding rate. We prove that C0 generically such systems exhibit one and at most a finite number of persistent periodic sinks attracting all the…
For a continuous map $f$ from the real line (half-open interval $[0,1)$) into itself let ent(f) denote the supremum of topological entropies of $f|_K$, where $K$ runs over all compact $f$-invariant subsets of $\mathbb{R}$ ($[0,1)$, respectively). It is proved that if $f$ is topologically transitive, then the best lower bound of ent(f) is $\log\sqrt{3}$ ($\log 3$, respectively) and it is not attained. This solves a problem posed by C{á}novas [Dyn. Syst. \textbf{24} (2009), no. 4, 473--483].
We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.
For a non-flat $C^3$ unimodal map with a Cantor attractor, we show that for any open cover $\mathcal U$ of this attractor, the complexity function $p(\mathcal U, n)$ is of order $n\log n$. In the appendix, we construct a non-renormalizable map with a Cantor attractor for which $p(\mathcal{U}, n)$ is bounded from above for any open cover $\mathcal{U}$.
We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\ep$-chains, and the minimal lengths of these $\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.
We prove that the flow generated by any interval map with zero topological entropy is minimally mean-attractable (MMA) and minimally mean-L-stable (MMLS). One of the consequences is that any oscillating sequence is linearly disjoint with all flows generated by interval maps with zero topological entropy. In particular, the Möbius function is orthogonal to all flows generated by interval maps with zero topological entropy (Sarnak's conjecture for interval maps). Another consequence is a non-trivial example of a flow having the discrete spectrum.
In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent and also invariant under topological conjugacy.
We study discontinuous interval maps generated by the action of erasing block substitutions on the binary expansion. After establishing some general properties of these maps, we categorize erasing block substitutions in a hierarchy of classes displaying progressively stronger erasing character. We investigate how this affects the dynamics of the corresponding interval maps, showing that the richest dynamical behavior (Devaney and Li-Yorke chaos, infinite topological entropy) is achieved at a precise step in this hierarchy, which we name completely erasing substitutions. KEYWORDS: Topological d…
We propose an inverse approach for dealing with interval maps based on the manner whereby their branches are related (folding property), instead of addressing the map equations as a whole. As a main result, we provide a symmetry-breaking framework for determining topological conjugacy of interval maps, a well-known open problem in ergodic theory. Implications thereof for the spectrum and eigenfunctions of the Perron-Frobenius operator are also discussed.
Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every $\varepsilon>0,$ there exist (complicate) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than $\varepsilon$ (simplicity). First we will show by means of three examples that for any graph that is not a tree the relatively simple maps (with small entropy) which are totally transitive (and hence robustly complicate) can be constructed so that t…
The topological entropy of a continuous self-map of a compact metric space can be defined in several distinct ways; when the space is not assumed compact, these definitions can lead to distinct invariants. The original, purely topological invariant defined by Adler et. al. is infinite if the system has at least one orbit with empty limit set. The Bowen-Dinaburg formulation (based on separated sets) is equivalent to the invariant defined by Friedland (based on a compactification of the inverse limit), and dominates another invariant proposed by Bowen (maximizing dispersion rates for orbits eman…
We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole with uniform expansion. We show that under suitable conditions the topological entropy varies locally Hoelder continuously with the local Hoelder exponent depending itself on the value of the topological entropy.
