Fractal metrics of Ruelle expanding maps and expanding ratios

In the theory of dynamical systems, it is well known that if f:X→Xf:X→X is a surjective equicontinuous map of a compactum X, then there is an admissible metric d for X   such that f:(X,d)→(X,d) is an isometry. In Reddy (1982) [12], Reddy proved that if f:X→X is a positively expansive map of a compactum X, then f   expands small distances. In this paper, we will study the similar properties of Ruelle expanding maps and admissible metrics. By use of the construction of the Alexandroff–Urysohn's metrization theorem we prove the following theorem which is a more precise result in case of Ruelle expanding maps (= positively expansive open maps): If f:X→Xf:X→X is a Ruelle expanding map of a compactum X   and any positive number s>1s>1, then there exist an admissible metric d for X   and positive numbers ϵ>0ϵ>0, λ   (1<λ1λ>1 such that if x,y∈Xx,y∈X, then d(f(x),f(y))=λd(x,y)d(f(x),f(y))=λd(x,y).