Restricted sensitivity, return time and entropy in Feldman–Katok and mean metrics

In this paper, by replacing the Bowen metric with the Feldman–Katok (FK) metric and the mean metric, respectively, we introduce measure-theoretic restricted FK (mean) sensitivities and topological restricted FK (mean) sensitivities. For a topological dynamical system, we discuss the relationships among measure-theoretic asymptotic FK rate with respect to sensitivity, topological asymptotic FK rate with respect to sensitivity, Brin–Katok local entropy and topological entropy. Parallel results are obtained for the mean metric case. In addition, we characterize the measure-theoretic entropy in terms of the exponential growth rate of the n-th return time to dynamical balls with respect to FK or mean metric. We also construct conditional entropy formulae with respect to FK metrics and the mean metrics.