Distance entropy of dynamical systems on noncompact-phase spaces

Let $X$ be a separable metric space not necessarily compact, and let $f: X\rightarrow X$ be a continuous transformation. From the viewpoint of Hausdorff dimension, the authors improve Bowen's method to introduce a dynamical quantity distance entropy, written as $ent_{H}(f;Y)$, for $f$ restricted on any given subset $Y$ of $X$; but it is essentially different from Bowen's entropy(1973). This quantity has some basic properties similar to Hausdorff dimension and is beneficial to estimating Hausdorff dimension of the dynamical system. The authors show that if $f$ is a local lipschitzian map with a lipschitzian constant $l$ then $ent_{H}(f;Y)\le\max\{0, $HD$(Y)\log l}$ for all $Y\subset X$; if $f$ is locally expanding with skewness $\lambda$ then $ent_{H}(f;Y)\ge $HD$(Y)\log\lambda$ for any $Y\subset X$. Here HD$(-)$ denotes the Hausdorff dimension. The countable stability of the distance entropy $ent_{H}$ proved in this paper, which generalizes the finite stability of Bowen's $h$-entropy (1971), implies that a continuous pointwise periodic map has the distance entropy zero. In addition, the authors show examples which demonstrate that this entropy describes the real complexity for dynamical systems over noncompact-phase space better than that of various other entropies.