56 research outputs found
On homeomorphisms with the two-sided limit shadowing property
We prove that the two-sided limit shadowing property is among the strongest
known notions of pseudo-orbit tracing. It implies shadowing, average shadowing,
asymptotic average shadowing and specification properties. We also introduce a
weaker notion that is called two-sided limit shadowing with a gap and prove
that it implies shadowing and transitivity. We show that those two properties
allow to characterize topological transitivity and mixing in a class of
expansive homeomorphisms and hence they characterize transitive (mixing) shifts
of finite type.Comment: 1 figure, comments are welcome
Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts
Positive topological entropy and distributional chaos are characterized for
hereditary shifts. A hereditary shift has positive topological entropy if and
only if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it is
not uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it is
not proximal (has more than one minimal set). As every spacing shift and every
beta shift is hereditary the results apply to those classes of shifts. Two open
problems on topological entropy and distributional chaos of spacing shifts from
an article of Banks et al. are solved thanks to this characterization.
Moreover, it is shown that a spacing shift has positive topological
entropy if and only if is a set of Poincar\'{e}
recurrence. Using a result of K\v{r}\'{\i}\v{z} an example of a proximal
spacing shift with positive entropy is constructed. Connections between spacing
shifts and difference sets are revealed and the methods of this paper are used
to obtain new proofs of some results on difference sets.Comment: Results contained in the paper were presented by the author at the
Visegrad Conference on Dynamical Systems, held in Bansk\'a Bystrica between
27 June and 3 July 2011, and at the 26th Summer Conference on Topology and
Its Applications hosted in July 26-29, 2011 by The City College of CUN
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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