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arXiv · 2209.13240

Minimal distance between random orbits

Sébastien Gouëzel·Jérôme Rousseau·Manuel Stadlbauer
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Abstract

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 R{\'e}nyi 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.

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Sébastien Gouëzel, Jérôme Rousseau, Manuel Stadlbauer. 2022-09-27. Minimal distance between random orbits. https://arxiv.org/abs/2209.13240

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