Development of concept of topological entropy for systems with multiple time

We study the topological entropy for dynamical systems with discrete or continuous multiple time. Due to the generalization of a well-known one time-dimensional result we show that the definition of topological entropy, using the approach for subshifts, leads to the zero entropy for many systems different from subshift. We define a new type of relative topological entropy to avoid this phenomenon. The generalization of Bowen’s power rule allows us to define topological and relative topological entropies for systems with continuous multiple time. As an application, we find a relation between the relative topological entropy and controllability of linear systems with continuous multiple time.     

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C ^∗-algebra.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.     

Dynamical intricacy and average sample complexity of amenable group actions

In 2018, Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of ?-action, based on the past works on the notion of intricacy in the research of brain network and probability theory. If one wants to take into account underlying system geometry in applications, more general group actions may need to be taken into consideration. In this paper, we consider this notion in the case of amenable group actions. We show that many basic properties in the ?-action case remain true. We also show that their suprema over covers or partitions are equal to the amenable topological entropy and the measure entropy, using the quasitiling technique in the theory of the amenable group.

     

On the mean square of the remainder for the euclidean lattice point counting problem

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.     

Mean topological dimension

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.     

Entropy points and applications

First notions of entropy point and uniform entropy point are introduced using Bowen's definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system.

Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.

     

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

Acta Mathematica Sinica, English Series - In this paper, we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group...     

Entropy,Chaos, and Weak Horseshoe for Infinite‐Dimensional Random Dynamical Systems

In this paper, we study the complicated dynamics of infinite‐dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we prove if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li‐Yorke. The complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system.© 2017 Wiley Periodicals, Inc.     

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.     

THE PROXIMAL RELATION,REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS

In this paper,we study the proximal relation,regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group acti...     

Switching systems and entropy

Switching systems are non-autonomous dynamical systems obtained by switching between two or more autonomous dynamical systems as time goes on. They can be mainly found in control theory, physics, economy, biomathematics, chaotic cryptography and of course in the theory of dynamical systems, in both discrete and continuous time. Much of the recent interest in these systems is related to the emergence of new properties by the mechanism of switching, a phenomenon known in the literature as Parrondo's paradox. In this paper we consider a discrete-time switching system composed of two affine transformations and show that the switched dynamics has the same topological entropy as the switching sequence. The complexity of the switching sequence, as measured by the topological entropy, is fully transferred, for example, to the switched dynamics in this particular case.     

Measuring complexity in a business cycle model of the Kaldor type

The purpose of this paper is to study the dynamical behavior of a family of two-dimensional nonlinear maps associated to an economic model. Our objective is to measure the complexity of the system using techniques of symbolic dynamics in order to compute the topological entropy. The analysis of the variation of this important topological invariant with the parameters of the system, allows us to distinguish different chaotic scenarios. Finally, we use a another topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy. This work provides an illustration of how our understanding of higher dimensional economic models can be enhanced by the theory of dynamical systems.     

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

Dynamics measured in a non-Archimedean field

We study dynamical systems using measures taking values in a non-Archimedean field. The underlying space for such measure is a zero-dimensional topological space. In this paper we elaborate on the natural translation of several notions, e.g., probability measures, isomorphic transformations, entropy, from classical dynamical systems to a non-Archimedean setting.     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

Density-equicontinuity and Density-sensitivity

In this paper we introduce the notions of (Banach) density-equicontinuity and densitysensitivity. On the equicontinuity side, it is shown that a topological dynamical system is densityequicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measuretheoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is densitysensitive for this measure.     

Markov shifts and topological entropy of families of homoclinic tangles

The existence of a homoclinic orbit in dynamical systems implies chaotic behaviour with positive entropy. In this work, we determine explicitly the Markov shifts associated to certain Smale horseshoe homoclinic orbits which allow us to compute a lower bound for the topological entropy that such a system can have. It is done associating a heteroclinic orbit which belongs to the same isotopy class and then determining the Markov partition of the dynamical core of an end periodic mapping.     

一类Smale空间上的C~*-代数自同构的熵

Ian Putnam利用Smale空间上的渐近等价关系定义了广群C~*-代数及其典则自同构.本文在零维Smale空间的情形下,计算此类C~*-自同构的逼近熵,证明了相应C~*-动力系统关于CNT熵和逼近熵的"变分原理"成立.由此推演出此类Smale空间上的Bowen测度诱导的C~*-代数上的态是此典则自同构的唯一平衡态.