  We investigate the so-called persistence problem of Sloane, exploiting
connections with the dynamics of circle maps and the ergodic theory of
$\mathbb{Z}^d$ actions. We also formulate a conjecture concerning the
asymptotic distribution of digits in long products of finitely many primes
whose truth would, in particular, solve the persistence problem. The heuristics
that we propose to complement our numerical studies can be thought in terms of
a simple model in statistical mechanics.
