Interval map, circle map, Lebesgue measure preserving, periodic points, Hausdorff dimension, upper box dimension, shadowing, generic properties

In this article we study dynamical behaviour of generic Lebesgue measure- preserving interval maps. We show that for each k greater or equal than 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 greater of equal than 1. Finally, we show that the generic continuous maps of the inter- val which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.

We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k>0 that the set of periodic points of period k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Furthermore, building on this result, we show that there is a dense collection of transitive Lebesgue measure preserving interval map whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k>0. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.