Spectral decomposition for rescaling expansive flows with rescaled shadowing

Abstract

In this paper, we introduce the concepts of rescaled expansiveness and the rescaled shadowing property for flows on metric spaces which are dynamical properties, and present a spectral decomposition theorem for flows. More precisely, we prove that if a flow is rescaling expansive and has the rescaled shadowing property on a locally compact metric space, then it admits the spectral decomposition. Moreover, we show that if a flow on locally compact metric space has the rescaled shadowing property then its restriction on nonwandering set also has the rescaled shadowing property.

Equivalent Condition of the Measure Shadowing Property on Metric Spaces

“…Next, we recall the definition of the shadowing property for a continuous flow on a metric space X in [11]. We begin by introducing the setting in metric space.…”

“…Regarding the dynamics of noncompact systems, a well-known difficulty is that some dynamical properties (i.e., properties that are invariant under conjugacy) on compact metric spaces may not be dynamical properties on noncompact metric spaces. For example, in [11], the authors provided an example to demonstrate that the classical shadowing property for flows on noncompact metric spaces depends on the choice of metrics; moreover, they introduced the notion of the shadowing property of continuous flows on metric spaces and showed that the extended definition is a dynamical property.…”

Flows with the weak two-sided limit shadowing property

“…The famous spectral decomposition theorem by Smale [22] says that the nonwandering set Ω(f ) of an Axiom A diffeomorphism f on a compact smooth manifold admits the spectral decomposition, i.e., Ω(f ) can be decomposed as a disjoint union of finite basic sets B, that is, each B is a compact invariant set such that f | B is topologically transitive. There are many works that generalize the Smale's spectral decomposition theorem to more general settings (e.g., see [1,10,11,12,14,16,18,19]). Komuro [14] proved the topological version of the Smale's spectral decomposition theorem for flows on compact metric spaces.…”