Some Consequences of the Shadowing Property in Low Dimensions

TL;DR

This paper explores the implications of the shadowing property in low-dimensional dynamical systems, revealing conditions under which periodic orbits exist and characterizing systems with certain shadowing regularities.

Contribution

It demonstrates that shadowing in two-dimensional systems guarantees periodic orbits in transitive classes and characterizes circle endomorphisms with Hölder shadowing as expanding maps.

Findings

Shadowing implies periodic orbits in 2D homeomorphisms.

Existence of a $C^$ Kupka-Smale diffeomorphism with shadowing and an aperiodic class.

Hb6lder shadowing with lpha>1/2 on circle endomorphisms implies conjugacy to an expanding map.

Abstract

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.