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Some consequences of the shadowing property in low dimensions

Published online by Cambridge University Press:  08 March 2013

ANDRES KOROPECKI  and
ENRIQUE R. PUJALS
ANDRES KOROPECKI
Affiliation:
Universidade Federal Fluminense, Instituto de Matemática, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brazil email ak@id.uff.br
ENRIQUE R. PUJALS
Affiliation:
IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil email enrique@impa.br
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Abstract

We consider low-dimensional systems with the shadowing property and we study the problem of existence of periodic orbits. 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 $-Hölder shadowing property with $\alpha \gt 1/ 2$ implies that the system is conjugate to an expanding map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1273 - 1309
Copyright
Copyright ©2013 Cambridge University Press 

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