Some consequences of the shadowing property in low dimensions

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.