The Brin–Katok formula for dynamical systems admitting mistakes

Abstract

Given a topological dynamical system (X,T), i.e., T is a continuous transformation on a compact metric space X), that admits mistakes and an invariant measure, this paper proves the Brin–Katok formula in this case. In particular, this paper finds that it suffices to study a mistake function which is linear in n and monotonic with respect to Δ. Consequently, this paper shows the Brin–Katok formula for the mean Bowen ball replacing the Bowen metrics with the mean Bowen metrics. The results presented in this paper may have some applications in the study of other properties of a dynamical system which admits small errors.