Rigidity of critical circle mappings, I

We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $\alpha$ depends only on the rate of convergence. We also give examples of $C^\infty$ critical circle maps with the same rotation number that are not $C^{1+\beta}$ conjugate for any $\beta>0$

On Sloane's persistence problem

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.Comment: 5 figure