Invariant measures for typical continuous maps on manifolds

Eleonora Catsigeras (1) , Serge Troubetzkoy (2)

1 IMERL - Instituto de Matemática y Estadística Rafael Laguardia [Montevideo]
2 I2M - Institut de Mathématiques de Marseille

We study the invariant measures of typical $C^0$ maps on compact connected manifolds with or without boundary, and also of typical homeomorphisms. We prove that the weak$^*$ closure of the set of ergodic measures coincides with the weak$^*$ closure of the set of measures supported on periodic orbits and also coincides with the set of pseudo-physical measures. Furthermore, we show that this set has empty interior in the set of invariant measures.

Domaines: Systèmes dynamiques [math.DS]

Eleonora Catsigeras, Serge Troubetzkoy. Invariant measures for typical continuous maps on manifolds. Nonlinearity, 2019, 10, pp.3981-4001. ⟨hal-01915783⟩