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.
Highlights
The theory of hyperbolic dynamical systems relies heavily on Anosov’s closing lemma ([1]) and the notion of specification as termed in Bowen’s work in 1971 ([7])
Senti and two of the present authors ([11]) used such structures to prove fluctuation laws of ergodic sums over periodic orbits, which can be expressed in terms of asymptotic normal distributions
In this paper we define the notion of strong measure expansive system and in Theorems B and C we prove that for a transitive and strong measure expansive homeomorphism, the notions of shadowing and specification are equivalent
Summary
The theory of hyperbolic dynamical systems relies heavily on Anosov’s closing lemma ([1]) and the notion of specification as termed in Bowen’s work in 1971 ([7]). While in the non-periodic case, we prove in Theorem A that shadowing and specification properties are equivalent, we found three different definitions of periodic shadowing in the literature (see [10], [13] and [17]). In this paper we define the notion of strong measure expansive system and in Theorems B and C we prove that for a transitive and strong measure expansive homeomorphism, the notions of shadowing and specification are equivalent. A homeomorphism f is called n-expansive if there is δ > 0 such that, for every x ∈ X, Γδ(x) has at most n elements
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