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 > 0, λ (1 < λ < s) such that if x, y ∈ X and d(x, y), 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:where dim H (X, d), D d (X) and D d (X) denote the Hausdorff dimension, the lower boxcounting 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)) = λd(x, y).
Let Γ be a group with subgroup Λ. We show, under certain conditions , that an almost periodic point for the action of Λ is also almost periodic for Γ. This is applied to obtain a theorem of Glasner.
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