We give examples of endomorphisms in dimension one with infinite topological entropy which are $α$-Hölder and $(1,p)$-Sobolev for all $0\leqα<1$ and $1\leq p<\infty$. This is constructed within a family of endomorphisms with infinite topological entropy and which traverse all $α$-Hölder and $(1,p)$-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, in some parametrized families we present sufficient conditions for the continuity of that entropy with respect to the parameter. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of t…
We study the jump of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^r([0; 1])$ with $h_{top}(f) > \frac{\log^+ \|f'\|_\infty}{r}$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.
In this note we construct measures of maximal entropy for a certain class of maps with critical points called Viana maps. The main ingredients of the proof are the non-uniform expansion features and the slow recurrence (to the critical set) of generic points with respect to the natural candidates for attaining the topological entropy.
We define a family of discontinuous maps on the circle, called Bowen-Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor-Thurston theory for one dimensional maps.
We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.
In this paper, we define the core entropy for postcritically-finite Newton maps and study its continuity within this family. We show that the entropy function is not continuous in this family, which is different from the polynomial case studied by Thurston, Gao, Dudko- Schleicher, Tiozzo [Th+, GT, DS, Ti2], and describe completely the continuity of the entropy function at generic parameters.
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.
We introduce two parametrized families of piecewise affine maps on $[0,1]^2$ and $[0,1]^3$, as generalizations of the heterochaos baker maps which were introduced and investigated in [Y. Saiki, H. Takahasi, J. A. Yorke, Nonlinearity, 34 (2021), 5744--5761] as minimal models of the unstable dimension variability in multidimensional dynamical systems. We show that natural coding spaces of these maps coincide with the Dyck system that has come from the theory of languages. Based on this coincidence, we start to develop a complementary analysis on their invariant measures. As a first attempt, we s…
We consider some smooth maps on a bouquet of circles. For these maps we can compute the number of fixed points, the existence of periodic points and an exact formula for topological entropy. We use Lefschetz fixed point theory and actions of our maps on both the fundamental group and the first homology group.
We prove that the topological entropy of real unimodal maps depends as a Hoelder continuous function of the kneading parameter, and the local Hoelder exponent equals, up to a factor log 2, the value of the function at that point.
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.
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.
In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.
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 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.
This work deals with a special case of family of birational maps f : C2 -> C2 dynamically classified in [9]. In this work we study the zero entropy sub families of f. The sequence of degrees dn associated to the iterates of f is found to grow periodically, linearly, quadratically or exponentially. Explicit invariant fibrations for zero entropy families and all the integrable and periodic mappings inside the family f are given.
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.
In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston \cite{MT} and for cubic maps by Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will prove the general case.
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 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.
In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Biś. Some examples with positive or zero Hausdorff metric entropy are given. Moreover, some notions of chaos are also well generalized for finitely generated semigroups from a set-valued view.
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.
Let $M$ be a compact real analytic manifold of finite dimension. There is a function $a: (0,+\infty)\to [0,+\infty)$ with $\lim_{t\to0}a(t)=0$ such that, the tail entropy $h^{*}(f,\varepsilon)$ of any real analytic map $f$ on $M$ is uniformly bounded above by the scale $a(\varepsilon)$.
The category of metric spaces is a subcategory of quasi-metric spaces. In this paper the notion of entropy for the continuous maps of a quasi-metric space is extended via spanning and separated sets. Moreover, two metric spaces that are associated to a given quasi-metric space are introduced and the entropy of a map of a given quasi-metric space and the maps of its associated metric spaces are compared. It is shown that the entropy of a map when symmetric properties is included is grater or equal to the entropy in the case that the symmetric property of the space is not considered.
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…
We consider a class $\mathcal{F}$ of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of $[0,1]^{\mathbb{Z}_+}$. For a multi-map in $\mathcal{F}$, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in $\mathcal{F}$.
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 construct measures of maximal $u$-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has a finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.
We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that $h_{\rm top}(f)>2\log\|f'\|_{\infty}/r$ possesses measures with maximal entropy.
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…
In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for se…
We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape $(+-+)$ which was posed in arXiv:1901.03458 based on experimental evidence.
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…
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.
This work dynamically classifies a 9-parametric family of birational maps f : C2 -> C2. From the sequence of the degrees dn of the iterates of f, we find the dynamical degree delta(f) of f. We identify when dn grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim in [4] as one of its subfamilies.
Given a closed, oriented, compact surface $S$ of constant negative curvature and genus $g \ge 2$, we study the measure-theoretic entropy of the Bowen-Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$-sided fundamental polygon of the surface $S$ and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of $S$ and prove the following flexibility result: the measure-theoretic entropy takes all values between $0$ and a maximum that is achieved on the surface…
We study the properties of $Φ$-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $Φ$-irregular set in terms of entropy on chain recurrent classes and prove that $Φ$-irregular sets of full entropy are typical. We also consider $Φ$-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measur…
For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the Hölder or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the $C^1$-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are Hölder and/or Sobolev of every exponent.
In this paper, we show that, under some technical assumptions, the Kolmogorov-Sinai entropy and the permutation entropy are equal for one-dimensional maps if there exists a countable partition of the domain of definition into intervals such that the considered map is monotone on each of those intervals. This is a generalization of a result by Bandt, Pompe and G. Keller, who showed that the above holds true under the additional assumptions that the number of intervals on which the map is monotone is finite and that the map is continuous on each of those intervals.
In \cite{Ch91a} it was shown that the billiard ball map for the periodic Lorentz gas has infinite topological entropy. In this article we study the set of points with infinite Lyapunov exponents. Using the cell structure developed in \cite{BSC90,Ku} we construct an ergodic invariant probability measure with infinite topological entropy supported on this set. Since the topological entropy is infinite this is a measure of maximal entropy. From the construction it is clear that there many such measures can coexist on a single component of topological transitivity. We also construct an ergodic inv…
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows wher…
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $\mathbb{Z}^k$-actions which are generated by random compositions of the generators of $\mathbb{Z}^k$-actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $\mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $\mathbb{Z}^k$(or $\mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on fi…
In this paper, various chaotic properties and their relationships for interval maps are discussed. It is shown that the proximal relation is an equivalence relation for any zero entropy interval map. The structure of the set of $f$-nonseparable pairs is well demonstrated and so is its relationship to Li-Yorke chaos. For a zero entropy interval map, it is shown that a pair is a sequence entropy pair if and only if it is $f$-nonseparable. Moreover, some equivalent conditions of positive entropy which relate to the number "3" are obtained. It is shown that for an interval map if it is topological…
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…
We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential…
