Fractal metrics of Ruelle expanding maps and expanding ratios

Abstract

In the theory of dynamical systems, it is well known that if f : 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 → X is a Ruelle expanding map of a compactum X and any positive number s > 1 , then there exist an admissible metric d for X and positive numbers ϵ > 0 , λ ( 1 < λ < s ) such that if x , y ∈ X and d ( x , y ) ⩽ ϵ , then d ( f ( x ) , f ( y ) ) = λ d ( x , y ) . For a case of graphs, we prove that if f : X → X is a positively expansive map of a graph X (= 1-dimensional compact polyhedron), then the same conclusion holds. In these cases, the metrics d satisfy the following equality: dim H ( X , d ) = D ̲ d ( X ) = D d ( X ) = h ( f ) log λ , where dim H ( X , d ) , D ̲ d ( X ) and D d ( X ) denote the Hausdorff dimension, the lower box-counting dimension and the upper box-counting dimension of the compact metric space ( X , d ) respectively, and h ( f ) is the topological entropy of f. This implies that such a metric d is a “fractal” metric for X. In fact, we can consider that the compact metric space ( X , d ) has some sort of local self-similarity with respect to the inverse f −1 of f and the similarity ratio 1 / λ . Also, we prove that if f : X → X is an expanding homeomorphism of a noncompact metric space X, then there exist an admissible metric d for X and a positive number λ > 1 such that if x , y ∈ X , then d ( f ( x ) , f ( y ) ) = λ d ( x , y ) .