Description:
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 € -transitive class, and in contrast we provide an example of a C1 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 α -Holder shadowing property with α > 1=2 implies that the system is conjugate to an expanding map.