A spectral decomposition for flows 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 on a compact smooth manifold can be written as a finite union of disjoint closed invariant sets on which 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, on a set is a collection of subsets of satisfying the following properties:
- (U1)
each member of contains the diagonal ,
- (U2)
if and , then ,
- (U3)
if and are members of , then is also a member of ,
- (U4)
if , then ,
- (U5)
for every there is such that , where
The set equipped with a uniformity is called a uniform space and an element of is called an entourage of . 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 be a flow on a uniform space . Given an entourage and , a -pseudo orbit is a pair of doubly infinite sequences such that and for all . For the sequence , we
Acknowledgments
This work was financially supported by research fund of Chungnam National University, Republic of Korea in 2016.
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