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

2021/03/26 by Jernej Činč, Jozef Bobok, Činč, Jernej +5
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.2103.14309

Several typos corrected and historical remarks updated

arxiv created 2021/04/09 · arxiv updated 2021/04/12

Abstract

We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k ≥ 1 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 ≥ 1. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.

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