Elsevier

Nonlinear Analysis

Volume 200, November 2020, 111982
Nonlinear Analysis

A spectral decomposition for flows on uniform spaces

https://doi.org/10.1016/j.na.2020.111982Get rights and content

Abstract

We study the behavior of flows on uniform spaces and consider expansivity, pseudo orbit tracing property and chain recurrence. The main result of this article is a generalization of spectral decomposition theorem to flows that are expansive and have the pseudo orbit tracing property on uniform spaces.

Introduction

In the study of dynamical systems, stability is one of the most important topics. The pseudo orbit tracing property(or shadowing property) plays a very important role in the theory of structural stability. Another important property is expansivity. The theories of pseudo orbit tracing property and expansivity are developed for discrete dynamical systems (see [3], [13], [14], [18], [19]). Expansive homeomorphisms with shadowing property are called topologically Anosov. One of the most important results of this theory is the spectral decomposition theorem, first obtained by Smale [16]. Spectral decomposition theorem says that the nonwandering set of an Anosov diffeomorphism f on a compact smooth manifold can be written as a finite union of disjoint closed invariant sets on which f is topologically transitive. It has been extended to homeomorphisms on compact metric spaces by Aoki [2]. Afterwards, Das et al. [6] have extended spectral decomposition theorem to homeomorphisms on noncompact and non-metrizable spaces.

Many of the dynamical results for homeomorphisms can be considered in continuous dynamical systems. Flows with expansive property are studied by Bowen and Walters [5]. Also, the results involving the expansivity and the pseudo orbit tracing property can be extended to the case of flows. The study of flows with the pseudo orbit tracing property has been done by Franke and Selgrade [7], and Thomas [17]. Spectral decomposition theorem has been extended to flows [15], [16]. In [10], Komuro discussed the spectral decomposition theorem for the case of expansive flows with finite pseudo orbit tracing property on compact metric spaces. There have been several results on the pseudo orbit tracing property and expansivity for flows (see [1], [4], [7], [12]).

In this paper, we introduce notions of pseudo orbit, pseudo orbit tracing property, and expansivity for flows on uniform spaces. And we extend spectral decomposition theorem to flows on compact uniform spaces. Moreover, we investigate the properties of nonwandering and chain recurrent sets for flows on uniform spaces.

Section snippets

Preliminaries

A uniform structure, or uniformity, U on a set X is a collection of subsets of X×X satisfying the following properties:

  • (U1)

    each member of U contains the diagonal X,

  • (U2)

    if DU and DEX×X, then EU,

  • (U3)

    if D and E are members of U, then DE is also a member of U,

  • (U4)

    if DU, then D1={(y,x):(x,y)D}U,

  • (U5)

    for every DU there is EU such that E2=EED, where EE={(x,y):zX with (x,z)E,(z,y)E}.

The set X equipped with a uniformity U is called a uniform space and an element of U is called an entourage of X. An

Spectral decomposition theorem

In this section, we prove a spectral decomposition theorem for expansive flow with the pseudo orbit tracing property on compact uniform spaces. We first introduce notions of pseudo orbit, pseudo orbit tracing property and chain recurrence for flows on uniform spaces.

Let ϕ be a flow on a uniform space X. Given an entourage DU and T>0, a (D,T)-pseudo orbit is a pair of doubly infinite sequences ({xi}i=,{ti}i=) such that tiT and (ϕti(xi),xi+1)D for all iZ. For the sequence {ti}i=, we

Acknowledgments

This work was financially supported by research fund of Chungnam National University, Republic of Korea in 2016.

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