In this paper, he interconnection of some regicide prepares between a continuous self map and is inverse limit is studied. It has been proved that (1) their invariant Borel probability measures are identical up to homoeomorphism and (2) they preserve uniform positive entropy property simultaneously. As application, it is also proved that the upper semi-continuous properties of their entropy maps are restricted each other, and he entropy map of he asymptotically h-expansive continuous map is upper semi-continuous, at he same item a continuous map having u.p.e. is topological weak-mixing.
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