2,153 research outputs found
Generic Points for Dynamical Systems with Average Shadowing
It is proved that to every invariant measure of a compact dynamical system
one can associate a certain asymptotic pseudo orbit such that any point
asymptotically tracing in average that pseudo orbit is generic for the measure.
It follows that the asymptotic average shadowing property implies that every
invariant measure has a generic point. The proof is based on the properties of
the Besicovitch pseudometric DB which are of independent interest. It is proved
among the other things that the set of generic points of ergodic measures is a
closed set with respect to DB. It is also showed that the weak specification
property implies the average asymptotic shadowing property thus the theory
presented generalizes most known results on the existence of generic points for
arbitrary invariant measures
Positive Entropy Through Pointwise Dynamics
We define some pointwise properties of topological dynamical systems and give
pointwise conditions for such a system possesses positive topological entropy.
We give sufficient conditions to obtain positive topological entropy for maps
which are approximated by maps with the shadowing property in a uniform way.Comment: 8 page
Variational equalities of entropy in nonuniformly hyperbolic systems
In this paper we prove that for an ergodic hyperbolic measure of a
diffeomorphism on a Riemannian manifold , there is an
-full measured set such that for every invariant
probability , the metric
entropy of is equal to the topological entropy of saturated set
consisting of generic points of :
Moreover, for every nonempty, compact and connected subset of
with the same hyperbolic rate, we
compute the topological entropy of saturated set of by the following
equality:
In particular these results can be applied (i) to the nonuniformy hyperbolic
diffeomorphisms described by Katok, (ii) to the robustly transitive partially
hyperbolic diffeomorphisms described by ~Ma{\~{n}}{\'{e}}, (iii) to the
robustly transitive non-partially hyperbolic diffeomorphisms described by
Bonatti-Viana. In all these cases
contains an open subset of .Comment: Transactions of the American Mathematical Society, to appear,see
http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06780-X
Hyperbolic periodic points for chain hyperbolic homoclinic classes
In this paper we establish a closing property and a hyperbolic closing
property for thin trapped chain hyperbolic homoclinic classes with one
dimensional center in partial hyperbolicity setting. Taking advantage of theses
properties, we prove that the growth rate of the number of hyperbolic periodic
points is equal to the topological entropy. We also obtain that the hyperbolic
periodic measures are dense in the space of invariant measures.Comment: 15 pages, 1 figure
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