126 research outputs found
On almost specification and average shadowing properties
In this paper we study relations between almost specification property,
asymptotic average shadowing property and average shadowing property for
dynamical systems on compact metric spaces. We show implications between these
properties and relate them to other important notions such as shadowing,
transitivity, invariant measures, etc. We provide examples that compactness is
a necessary condition for these implications to hold. As a consequence of our
methodology we also obtain a proof that limit shadowing in chain transitive
systems implies shadowing.Comment: 2 figure
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
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