Li-Yorke Pairs Research Articles

Some properties of dynamical systems having shadowing

Let X be a compact metric space and f:X→X be a continuous map having the shadowing property. In this paper, first, we prove that if X is connected, then ωf is continuous if, and only if, f is minimal. Second, we prove that the continuity of ωf yield CR(f)=R(f)=AP(f). Third, it is shown that if any point is recurrent then the collection of minimal sets is closed in (2X,dH). Finally, we show that the existence of a Li-Yorke pair is sufficient to have positive topological sequence entropy and if furthermore, f has s-shadowing property then f is Li-Yorke chaotic.

Dynamics of monotone maps on regular curves

Let X be a regular curve and let f:X→X be a monotone map. In this paper, the nonwandering set of f and the structure of ω-limit (resp. α-limit) sets for f are investigated. Denote by Per(f) (resp. Ω(f)) the set of periodic points (resp. nonwandering points) of f. We show that whenever the set Per(f) is empty, then f admits a unique minimal set which is totally minimal and either a circle or a Cantor set. If Per(f) is nonempty, we show that Ω(f)=Per(f)‾. In addition, every infinite minimal set is a Cantor set which is an adding machine. On the other hand, we show that f has no Li-Yorke pairs, in particular it has zero entropy. Further results related to the equicontinuity are also obtained.

Packing Entropy of Saturated Sets for Nonuniformly Hyperbolic Systems

In this paper, we prove that for a nonuniformly hyperbolic system [Formula: see text], and for every convex and compact subset [Formula: see text] with the same hyperbolic rate, the packing entropy of saturated set [Formula: see text] equals the supremum of the metric entropy of all measures in K. Here [Formula: see text] denotes the set of all [Formula: see text]-invariant Borel probability measures supported on [Formula: see text]. As an application, we describe the size of the set of mean Li–Yorke pairs.

Chaotic sets and Hausdorff dimension for Lüroth expansions

We provide new similarities between regular continued fractions and Lüroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the Lüroth transformation of results by W. Liu, B. Li [18] and W. Liu, S. Wang [19] on the distal, asymptotic and Li-Yorke pairs for the Gauss map.

Entropy of induced maps of regular curves homeomorphisms

Let f : X → X be a self homeomorphism of a continuum X, we show that the topological entropy of the induced system (2X,2f) is infinite provided that X\\Ω(f) is not empty. If furthermore X is a regular curve then it is shown that (2X,2f) has infinite topological entropy if and only if X\\Ω(f) is not empty. Moreover we prove for the induced system (C(X),C(f)) the equivalence between the following properties: (i) zero topological entropy; (ii) there is no Li-Yorke pair and (iii) for any periodic subcontinnum A of X and any connected component C of X\\Ω(f), C ⊂ A if A ∩ C ≠ ∅. In particular, the topological entropy of either (2X,2f) or (C(X),C(f)) has only two possible values 0 or ∞. At the end, we give an example of a pointwise periodic rational curve homeomorphism F : Y → Y with infinite topological entropy induced map C(F).

Investigating Distributional Chaos for Operators on Fréchet Spaces

In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.

Homeomorphism of Hereditarily Locally Connected Continua

Let $$f:X\rightarrow X$$ be a hereditarily locally connected continuum homeomorphism and denote respectively by P(f), AP(f) and $$\Omega (f)$$ , the sets of periodic points, almost periodic points and the non-wandering points of f. We show that any $$\omega $$ -limit set (resp. $$\alpha $$ -limit set) is minimal. Moreover, we show that $$\Omega (f)=AP(f)$$ . We also prove that if $$P(f)=\emptyset $$ , then there exists a unique minimal set. On the other hand, if $$P(f)\ne \emptyset $$ then we prove that any infinite minimal set has the adding machine structure and the absence of Li-Yorke pairs. Consequently, we partially solve the positive entropy conjecture which remains open even in the case of hereditarily locally connected continuum.

On regular curve homeomorphisms without periodic points

The purpose of this paper is to study the dynamics of regular curve homeomorphisms without periodic points. We prove mainly that they are extensions of irrational rotations of the circle via a monotone factor map collapsing all proximal pairs and we prove also the absence of Li-Yorke pairs. Furthermore, we give a full characterization of minimal sets. In particular, we get that the circle is the only regular curve supporting a minimal (or a transitive) -action. At the end of the paper, we give some counter-examples on rational curves.

Dynamics of Induced Maps on the Space of Probability Measures

For a continuous self-map f on a compact metric space X, we provide two simple examples: the first confirms that shadowing of (X, f) is not inherited by $$({\mathcal {M}}(X),{\hat{f}})$$ in general, and the other satisfies that both (X, f) and $$({\mathcal {M}}(X),{\hat{f}})$$ have no Li–Yorke pair, where $${\mathcal {M}}(X)$$ be the space of all Borel probability measures on X. Then we prove that chain transitivity of (X, f) implies chain mixing of $$({\mathcal {M}}(X),{\hat{f}})$$ , and provide an example to deny the converse. For a non-autonomous system $$(X,f_{0,\infty })$$ , we prove that weak mixing of $$({\mathcal {M}}(X),\hat{f}_{0,\infty })$$ implies that of $$(X,f_{0,\infty })$$ , and give an example to deny the converse, where $$f_{0,\infty }=\{f_n\}_{n=0}^\infty $$ is a sequence of continuous self-maps on X. We also prove that if $$f_n$$ is surjective for all $$n\ge 0$$ , then chain mixing of $$({\mathcal {M}}(X),{\hat{f}}_{0,\infty })$$ always holds true, and shadowing of $$({\mathcal {M}}(X),{\hat{f}}_{0,\infty })$$ implies mixing of $$(X, f_{0,\infty })$$ . If $$X=I$$ is an interval, we obtain a sharp condition such that transitivity is equivalent between (I, f) and $$({\mathcal {M}}(I),{\hat{f}})$$ . Although $$({\mathcal {M}}(I),{\hat{f}})$$ has infinite topological entropy for any transitive system (I, f), we give an example such that $$(I,f_{0,\infty })$$ is transitive but $$({\mathcal {M}}(I),{\hat{f}}_{0,\infty })$$ has zero topological entropy.

Equicontinuity and Li–Yorke pairs of dendrite maps

ABSTRACT In this paper, relationships between equicontinuity of a dendrite map f on , absence of Li–Yorke pairs, collection of minimal sets and regularly recurrent points are investigated.

The topological entropy of endomorphisms of Lie groups

In this paper, we determine the topological entropy h(φ) of a continuous endomorphism φ of a Lie group G. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when G is noncompact, the well known Bowen’s formula for the entropy hd(φ) associated to a left invariant distance d just provides an upper bound to h(φ), which is characterized by the so called variational principle. We prove that $$h(\phi ) = h(\phi {|_{T({G_\phi })}})$$ where Gφ is the maximal connected subgroup of G such that φ(Gφ) = Gφ, and T(Gφ) is the maximal torus in the center of Gφ. This result shows that the computation of the topological entropy of a continuous endomorphism of a Lie group reduces to the classical formula for the topological entropy of a continuous endomorphism of a torus. Our approach explores the relation between null topological entropy and the nonexistence of Li-Yorke pairs and also relies strongly on the structure theory of Lie groups.

Chaos theory from the mathematical point of view

This work is intended as an attempt to survey existingde finitions of chaos for discrete dynamical systems. Discussion is restricted to the settingof topological dynamics, while the measure-theoretic (ergodic theory) and smooth (differentiable dynamical systems) aspects are omitted as exceedingt he scope of this paper. Chaos theory is understood here as a part of topological dynamics, so aforementioned definitions of chaos are just examples of particular dynamical system properties, and are considered inside the framework of the mathematical theory of discrete dynamical systems. It is not the purpose of this article to study chaos theory understood as a new kind of interdisciplinary branch of science devoted to nonlinear phenomena. As for prerequisites, the reader is expected to possess some mathematical maturity, and to be familiar with basic topology of (compact) metric spaces. No preliminary knowledge of the dynamical systems theory is required, however some is recommended. The first two section are devoted to general discussion of the term „chaos” and contains authors opinion on this subject. To facilitate access to the rest of the article some relevant material from the dynamical system theory is briefly repeated in the third section. The next section (Section 4) introduces the notion of topological transitivity along with some stronger variants, namely topological mixing and weak mixing. Section 5 gives a detailed account of the famous Sharkovskii’s Theorem in its full generality. This is required for characterization of chaotic interval maps. Sections 6-13 are devoted to various notions of chaos or related to chaos in dynamical systems. Each section contains an attempt to motivate the notion, historical background and formal definition followed with a review of known properties, relations between various notions of chaos, and some relevant open problems. Section 6 is devoted to a sensitivity to initial conditions – a notion which is accepted as a basic indicator of chaotic behavior. Section 7 introduces a definition of chaos accordingt o Auslander and Yorke. Section 8 examines the notion of Li-Yorke pair and Li-Yorke chaos. Section 9 deals with the definition of chaos introduced in Devaney’s book (Devaney chaos). Section 10 recalls some facts connected with symbolic dynamics, which provides a rich source of examples for various interestingb ehavior, and it is an indispensable tool for exploration of many systems. Section 11 describes the so-called “topological horseshoes”, which are generalizations of the famous example due to Smale. The existence of a horseshoe in a given dynamical system proves the existence of a subsystem with a dynamics similar to some symbolic dynamical system, hence with a very complicated behavior. Section 12 gives a brief exposition of the topological entropy and its relation to chaos. The review of various notions of chaos ends with section 13, containingd escription of distributional chaos.

Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems

In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems. We prove that Li-Yorke pairs of A-coupled-expanding system under some conditions have full Hausdorff dimension in the invariant set. Moreover we give a generalization of the result of [5] which is on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and have a self-similar invariant set, to the case of the dynamical systems topologically semi-conjugate to some kinds of subshifts. Furthermore Hausdorff dimension of “chaotic invariant set” for some kinds of A-coupled-expanding maps is shown.

Li-Yorke chaos for dendrite maps with zero topological entropy and <i>ω</i>-limit sets

Let $X$ be a dendrite with set of endpoints $E(X)$ closed and let $f:~X \to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $\omega$-limit set of $f$ then $L\cap P(f)\subset E(X)^{\prime}$, where $E(X)^{\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and $L$ is uncountable then $L\cap P(f)=\emptyset$. We also show that if $E(X)^{\prime}$ is finite then any uncountable $\omega$-limit set of $f$ has a decomposition and as a consequence if $f$ has a Li-Yorke pair $(x,y)$ with $\omega_f(x)$ or $\omega_f(y)$ is uncountable then $f$ is Li-Yorke chaotic.

On the Weakest Version of Distributional Chaos

The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li–Yorke pair. However, the definition can be strengthened to get DC[Formula: see text] which is a topological invariant and implies Li–Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC[Formula: see text] systems must have zero topological entropy.

Invariant Sets for Monotone Local Dendrite Maps

Let [Formula: see text] be a local dendrite and let [Formula: see text] be a monotone map. Denote by [Formula: see text], RR[Formula: see text], UR[Formula: see text], [Formula: see text] the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and [Formula: see text] the union of all [Formula: see text]-limit sets of [Formula: see text]. We show that if [Formula: see text] is nonempty, then (i) [Formula: see text]. (ii) R[Formula: see text] if and only if every cut point is a periodic point. If [Formula: see text] is empty, then (iii) [Formula: see text]. (iv) R[Formula: see text] if and only if [Formula: see text] is a circle and [Formula: see text] is topologically conjugate to an irrational rotation of the unit circle [Formula: see text]. On the other hand, we prove that [Formula: see text] has no Li–Yorke pair. Moreover, we show that the family of all [Formula: see text]-limit sets of [Formula: see text] is closed with respect to the Hausdorff metric.

Intra-orbit separation of dense orbits of dendrite maps

In this note, we will generalize the notion of separation index, which was introduced by Manjunath et al. (2006), to dendrite maps, and use the notion to characterize the intra-orbit separation for the orbits of continuous transitive dendrite maps. We will show: (i) For a dendrite map f, the separation index γ is greater than zero if and only if the set of fixed points of f is not an arc connected subspace. (ii) If the separation index γ of a dendrite map f is greater than zero, then for every 0<τ<γ and any pair of distinct points x and y on a dense orbit, {x,y} is a Li–Yorke pair of modulus τ.

Graph theoretic structure of maps of the Cantor space

In this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of spaces H({0,1}N) and C({0,1}N), the space of homeomorphisms and the space of self-maps of the Cantor space, respectively. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the comeager conjugacy class of the space H({0,1}N). The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li–Yorke pair, implying the well known Glasner–Weiss result that there is a comeager subset of H({0,1}N) each element of which has topological entropy zero. Our analogous investigation in C({0,1}N) yields a surprising result: there is a comeager subset of C({0,1}N) such that any two elements of this set are conjugate to each other by an element of H({0,1}N). Our description of this class also yields many old and new results concerning dynamics of a comeager subset of C({0,1}N).

DYNAMICS OF MONOTONE GRAPH, DENDRITE AND DENDROID MAPS

We show that, for monotone graph map f, all the ω-limit sets are finite whenever f has periodic point and for monotone dendrite map, any infinite ω-limit set does not contain periodic points. As a consequence, monotone graph and dendrite maps have no Li–Yorke pairs. However, we built a homeomorphism on a dendroid with a scrambled set having nonempty interior.

On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps

We investigate the prevalence of Li-Yorke pairs for C 2 and C 3 multimodal maps f with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If f is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally f admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for f × f. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where f has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.