Minimal distance between random orbits

Sébastien Gouëzel (1) , Jérôme Rousseau (2, 3) , Manuel Stadlbauer (4)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CREC - Centre de Recherche des Ecoles de St-Cyr Coëtquidan
3 UFBA - Universidade Federal da Bahia
4 UFRJ - Universidade Federal do Rio de Janeiro [Brasil] = Federal University of Rio de Janeiro [Brazil] = Université fédérale de Rio de Janeiro [Brésil]

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

Domaines: Systèmes dynamiques [math.DS]

Sébastien Gouëzel, Jérôme Rousseau, Manuel Stadlbauer. Minimal distance between random orbits. 2022. ⟨hal-03788538⟩