  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 H\"{o}lder potential, we prove the asymptotic length of the
longest common substring for a typical point converges and the limit depends on
the R\'{e}nyi 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.
