On specification and measure expansiveness

Abstract

We relate the local specification and periodic shadowing properties. We also clarify the relation between local weak specification and local specification if the system is measure expansive. The notion of strong measure expansiveness is introduced, and an example of a non-expansive systems with the strong measure expansive property is given. Moreover, we find a family of examples with the N -expansive property, which are not strong measure expansive. We finally show a spectral decomposition theorem for strong measure expansive dynamical systems with shadowing.

Positively N-expansive Homeomorphisms and the L-shadowing Property

“…To prove the shadowing property, let ε > 0 be given and consider δ > 0 such that every δ-pseudo orbit, that is also a two-sided limit pseudo-orbit, is both εshadowed and two-sided limit shadowed. It is known that the shadowing property is equivalent to the finite shadowing property (see [9] for a proof). Then it is sufficient to prove that any finite δ-pseudo orbit is ε-shadowed.…”

Topological stability and spectral decomposition for homeomorphisms on noncompact spaces

“…This shows that f is not n-expansive for any n ∈ N. Consequently, we cannot apply Theorem B in [2] directly to prove the spectral decomposition theorem in [4].…”

“…Recently, Cordeiro et. al [4] proved a spectral decomposition theorem for strong measure expansive homeomorphims with the shadowing property on compact metric spaces. Unfortunately, the proof in [4] contains a small gap in which can be revised by a slightly modified technique.…”

“…Recently, Cordeiro et. al gave a spectral decomposition for strong measure expansive homeomorphisms with the shadowing property on compact metric spaces (for more details, see Theorem D in [4]). Unfortunately, there is a small gap in the proof of the result.…”

Spectral decomposition for rescaling expansive flows with rescaled shadowing

“…Recently, there are many works that generalize the Smale's spectral decomposition theorem to more general settings. In particular, Lee et al [6,11] proved a spectral decomposition theorem for expansive homeomorphisms with the shadowing property on noncompact spaces, and Cordeiro et al [5] present a measurable version of spectral decomposition theorem for strongly measure expansive homeomorphisms with the shadowing property on compact metric spaces.…”