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Some consequences of the shadowing property in low dimensions
Published online by Cambridge University Press: 08 March 2013
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.
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- Copyright ©2013 Cambridge University Press
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