Lower bounds of the topological entropy for nonautonomous dynamical systems

In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.

CHAIN RECURRENT POINTS AND TOPOLOGICAL ENTROPY OF A TREE MAP

Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced.It is proved that: (1) f has zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f s has a division; (2) If f has zero topological entropy,then for any x∈CR(f)-P(f) the ω-limit set of x is an infinite minimal set.

The Topological Entropy Mechanism of Coronavirus Disease 2019 (COVID-19)

The biological principal or its detailed mechanism for the pandemic coronavirus disease 2019 (COVID-19) has been investigated and analyzed from the topological entropy approach. The findings thus obtained have provided very useful clues and information for developing both powerful and safe vaccines against the pandemic COVID-19.

TOPOLOGICAL ENTROPY OF PERIODIC COVEN CELLULAR AUTOMATA

We investigate topological entropy of periodic Coven cellular automatas; that is, the maps Fs： （0, 1）^z → {0, 1）^z defined by FB（x）i=xi＋^rПj=1（xi＋j＋bj）（mod 2）, where B = b1b2…br ∈ {0, 1}^r（r≥2）, is a periodic word. In particular, we prove that if the minimal period of B is greater than 5, the topological entropy is log 2.

A CHARACTERIZATION OF THE SOLUTIONS OF p-ORDER FEIGENBAUM'S FUNCTIONAL EQUATION WITH TOPOLOGICAL ENTROPY

Let f be a continuous anti-unimodal solution of a p-order Frigenbaum's dunctioal equation.A criterion is given to determine whether or not the topological entropy of j is zero.And a continuous anti-unimodal solution of 4-order equation with positive topological eotrpy is constructed.

TOPOLOGICAL ENTROPY AND IRREGULAR RECURRENCE

This paper is devoted to problems stated by Z. Zhou and ELi in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map f and its topological entropy. The negative answer follows by our recent paper. But for continuous maps of the interval and other more general one-dimensional spaces we give more results; in some cases the answer is positive.

A Note on Topological Entropy of Continuous Self-Maps

Topological entropy can be an indicator of complicated behavior in dynamical systems. It is first introduce by Adler, Konheim and McAndrew by using open covers in 1965. After that it is still an active research by many researchers to produce more properties and applications up to nowadays. The purpose of this paper is to review and explain most important concepts and results of topological entropies of continuous self-maps for dynamical systems on compact and non-compact topological and metric spaces. We give proofs for some of its elementary properties of the topological entropy. Slight modification on Adler＇s topological entropy is also presented.

Two Notes on Lower Bounds of Topological Entropy

Let f:XX be a selfmap of a compact connected polyhedron, using Nielsen fixed point theory, we give a better lower bound for topological entropy h(f) of f . In addition, if f:T mT m is a selfmap of m? 玹rous, some conditions under which log N ∞(f) is a best lower bound for

Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy

A class of minimal subshifts which display Schweizer-Smital chaos and have zero topotogical entropy is constructed, and it is proved that for a compact system, the positive topological entropy is not generally equivalent to Schweizer-Smital chaos occurring on the measure centre.

Topological Entropy of a Graph Map

Let G be a graph and f： G → G be a continuous map. Denote by h（f）, P（f）, AP（f）, R（f） and w（x, f） the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the w-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent： （1） h（f） 〉 O. （2） There exists an x ∈ G such that w（x, f） ∩ P（f） ≠θ and w（x, f） is an infinite set. （3） There exists an x ∈ G such that w（x, f） contains two minimal sets. （4） There exist x, y ∈G such that w（x, f） - w（y, f） is an uncountable set andw（y,f）∩w（x,f）≠θ. （5） There exist anx C Gand a closed subset A w（x,f） with f（A） A such that w（x,f） - A is an uncountable set. （6） R（f） - nP（f） ≠θ. （7） f｜P（f） is not pointwise equicontinuous.

The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth

In this article,we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth.We prove that on each energy level E(x,v)=k with k>c(L),where c(L)is Mane’s critical value,the EulerLagrange flow has positive topological entropy.This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.

Topological Entropy of a Class of Subshifts of Finite Type

In this paper, we construct a special class of subshifts of finite type. By studying the spectral radius of the transfer matrix associated with the subshift of finite type, we obtain an estimation of its topological entropy. Interestingly, we find that the topological entropy of this class of subshifts of finite type converges monotonically to log（n ＋ 1） （a constant only depends on the structure of the transfer matrices） as the increasing of the order of the transfer matrices.

On Local Constancy of Topological Entropy for Certain Partially Hyperbolic Diffeomorphisms

Let f ： M → M be a partially hyperbolic diffeomorphism on a closed Riemannian manifold with uniformly compact center foliation. We show that if the center foliation of f is of dimension one then the topological entropy is constant on a small C1 neighborhood of f.

ON THE ENTROPY OF FLOWS WITH REPARAMETRIZED GLUING ORBIT PROPERTY

We show that a flow or a semiflow with a weak form of reparametrized gluing orbit property has positive topological entropy if it is not minimal.

Upper Estimates on the Higher-dimensional Multifractal Spectrum of Local Entropy

We discuss the problem of higher-dimensional multifractal spectrum of local entropy for arbitrary invariant measures. By utilizing characteristics of a dynamical system, namely, higher-dimensional entropy capacities and higher-dimensional correlation entropies, we obtain three upper estimates on the hlgher-dimensional multifractal spectrum of local entropies. We also study the domain of higher-dimensional multifractai spetrum of entropies.

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T ： X → X be a uniformly continuous homeomorphism on a non-compact metric space （X, d）. Denote by X＊ = X ∪ {x＊} the one point compactification of X and T ＊ ： X＊ → X＊ the homeomorphism on X＊ satisfying T ＊｜X = T and T ＊x＊ = x＊. We show that their topological entropies satisfy hd（T, X） ≥ h（T ＊, X＊） if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.

A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic

Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.

The Multi-sensitivity and Topological Sequence Entropy of Dynamical System with Group Action

In this paper,we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group action.Furthermore,we also discuss the consistency of multi-sensitivity of a dynamical system(G■X)and its hyperspace dynamical system G■K(X).Moreover,we research the relationship between the multi-sensitivity of two dynamical systems and the multi-sensitivity of their product space dynamical system.Finally,we prove that if the topological sequence entropy of G■X vanishes,then so does that of its induced system G■M(X);if the topological sequence entropy of G■X is positive,then that of its induced system G■M(X)is infinity.

A Note on a Misuirewicz's Theorem

In this paper,we improve a Misuirewicz's theorem and give a new proof of small conjecture on topological entropy.

On Some Questions of Barge and Toledo

Barge asked if each homeomorphism of a hereditarily decomposable chainable continuum has zero topological entropy and Toledo asked if a homeomorphism of a chainable continuum can always be induced by square commuting diagram on inverse systems of finite graphs. We show in this note that if Toledos question has a positive answer then Barges question also has a positive answer.