Minimal distance between random orbits

We study the minimal distance between two orbit segments of length $n$, in
a random dynamical system with sufficiently good mixing properties. For the annealed version of this problem, the asymptotic behavior is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension. 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.

Joint work with Sébastien Gou ëzel and Manuel Stadlbauer.