Shadowing by non-uniformly hyperbolic periodic points and uniform hyperbolicity

Abstract

Abstract. We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.

On C1-class local diffeomorphisms whose periodic points are nonuniformly expanding

“…However, Mañé's theorem does not apply directly to our situation studied here, since f is not a diffeomorphism. In addition, the statement (1) of Theorem 1 is proved by Castro, Oliveira and Pinheiro [8] in the special case where f possesses the closing by periodic orbits property, and by Sun and Tian [28] in the generic case.…”

“…There have been a few results concerning this. One of these results is the remarkable Theorem A of Mañé [24] for C 1+Hölder endomorphisms of the unit circle T 1 ; some interesting other results for C 1 -class local diffeomorphisms of M n where n ≥ 2, have appeared in several recent papers [1,6,7,8,10].…”

On specification and measure expansiveness

“…Both notions can be formulated in terms of approximating orbits being periodic or not periodic. 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]). It turns out that for expansive transformations one derives the existence of periodic points leading to equivalent statements for periodic shadowing and specification.…”

Average Shadowing Property With Non Uniformly Hyperbolicity on Periodic Points

“…The hyperbolicity theory has many applications within other theory in dynamical systems such as chaos theory and structural stability. For example, A. Castro and et al in [19] obtained by nonuniformly hyperbolic periodic points, the uniform hyperbolicity and the shadowing property had an important rule in their theorems. In fact, it is the theory of nonuniformly hyperbolic dynamical systems that in applications provides the mathematical foundation of the theory of chaos.…”