  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.

Dynamical Systems