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

Jozef Bobok (1) , Jernej Činč (2) , Piotr Oprocha (3) , Serge Troubetzkoy (4)

1 CTU - Czech Technical University in Prague
2 University of Vienna [Vienna]
3 AGH UST - AGH University of Science and Technology [Krakow, PL]
4 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; 37E05; 37A05; 37B65; 37C20

Domaines: Systèmes dynamiques [math.DS]

Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy. PERIODIC POINTS AND SHADOWING PROPERTY FOR GENERIC LEBESGUE MEASURE PRESERVING INTERVAL MAPS. Nonlinearity, 2022, 35 (5), pp.2535-2558. ⟨10.1088/1361-6544/ac62df⟩. ⟨hal-03181370v3⟩