Closest Distance between Iterates of Typical Points

The shortest distance between the first $n$ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi entropy. For interval maps with the Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point.