Positive topological entropy for monotone recurrence relations

Abstract

We associate the topological entropy of monotone recurrence relations with the Aubry–Mather theory. If there exists an interval$[{\it\rho}_{0},{\it\rho}_{1}]$such that, for each${\it\omega}\in ({\it\rho}_{0},{\it\rho}_{1})$, all Birkhoff minimizers with rotation number${\it\omega}$do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.