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Pre-image entropy for maps on noncompact topological spaces
Received: August 24, 2012; Accepted: April 16, 2013
Abstract. We propose a new definition of pre-image entropy for continuous maps on noncompact topological spaces, investigate fundamental properties of the new pre-image entropy, and compare the new pre-image entropy with the existing ones. The defined 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.
Keywords: Pre-image entropy; Locally compact space; Alexandroff compactification; Hyperspace dynamical system.
AMS Subject classification: Primary: 54H20, 28D20
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