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Pre-image entropy for maps on noncompact topological spaces

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Author(s): Lei Liu

Journal: Acta Mathematica Universitatis Comenianae
ISSN 0862-9544

Volume: LXXXII;
Issue: 2;
Start page: 219;
Date: 2013;
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Keywords: Pre-image entropy | Locally compact space | Alexandroff compactification | hyperspace dynamical system

ABSTRACT
We propose a new definition of pre-imageentropy for continuous maps on noncompact topological spaces,investigate fundamental properties of the new pre-image entropy, andcompare the new pre-image entropy with the existing ones. Thedefined pre-image entropy generates that of Cheng and Newhouse. Yet, it holds various basic properties of Cheng and Newhouse's pre-image entropy, for example, the pre-image entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have the same pre-image entropy, the pre-image entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new pre-image entropy coincides with Cheng and Newhouse's pre-image entropy for compact systems.
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