Consequences of the Shadowing Property

Details Consequences of the Shadowing Property Consequences of the Shadowing Property

2011-09-23

arxiv.org/abs/1109.5074v1

Mathematics

Dynamical Systems

31 pages, 9 figures

Scientific paper

We consider low-dimensional systems with the shadowing property. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $\epsilon$-transitive class, and in contrast we provide an example of a $C^\infty$ Kupka-Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally we consider the case of transitive endomorphisms of the circle, and we prove that the $\alpha$-H\"older shadowing property with $\alpha>1/2$ implies that the system is conjugate to an expanding map.

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