Pre-image entropy for maps on noncompact topological spaces (Q2850102)

Topological pre-image entropy for a continuous self-map \(f\) of a metrizable, in general noncompact, topological space \(X\) is introduced. If there is no nonempty compact \(f\)-invariant subset \(F\) of \(X\), the preimage entropy of \(f\) is defined to be zero. Otherwise, for every such \(F\), one computes the pre-image entropy in the sense of \textit{W-Ch. Cheng} and \textit{Sh. E. Newhouse} [Ergodic Theory Dyn. Syst. 25, 1091--1113 (2005; Zbl 1098.37012)] for the restriction of the map \(f\) to the set \(F\) and then the preimage entropy of \(f\) is defined to be the supremum of the obtained values over all such sets \(F\). If \(X\) is compact then the pre-image entropy of \(f\) defined in such a way of course coincides with that in the sense of Cheng and Newhouse [loc. cit.]. Further, the pre-image entropy does not depend on the choice of a metric on the space \(X\) and it is an invariant of topological conjugacy. The author also proves some other properties of it.