PERIODIC POINTS AND SHADOWING PROPERTY FOR GENERIC LEBESGUE MEASURE PRESERVING INTERVAL MAPS

Jernej Činč , Jozef Bobok , Piotr Oprocha , Serge Troubetzkoy (1)

1 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: Interval map; circle map; Lebesgue measure preserving; periodic points; Hausdorff dimension; upper box dimension; shadowing; generic properties; April 9; 2021. 2020 Mathematics Subject Classification. 37E05; 37A05; 37B65; 37C20 Interval map

Domaines: Systèmes dynamiques [math.DS]

Jernej Činč, Jozef Bobok, Piotr Oprocha, Serge Troubetzkoy. PERIODIC POINTS AND SHADOWING PROPERTY FOR GENERIC LEBESGUE MEASURE PRESERVING INTERVAL MAPS. 2021. ⟨hal-03181370v2⟩