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Abstract
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.
Mathematics Subject Classification: Primary: 37B40, 37C45; Secondary: 37B10.
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