Periodic points and shadowing for generic Lebesgue measure-preserving interval maps

Abstract

In this article we study dynamical behaviour of generic Lebesgue measure-preserving interval maps. We show that for each k ⩾ 1 the set of periodic points of period at least k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Moreover, we obtain analogous results also in the context of generic Lebesgue measure-preserving circle maps. Furthermore, building on the former results, we show that there is a dense collection of transitive Lebesgue measure-preserving interval maps whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k ⩾ 1. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.