Irregular recurrence in compact metric spaces

For a continuous map f:X→Xf:X→X of a compact metric space, the set IR(f)IR(f) of irregularly recurrent points is the set of points which are upper density recurrent, but not lower density recurrent. These notions are related to the structure of the measure center, but many problems still remain open. We solve some of them. The main result, based on examples by Obadalová and Smítal [Obadalová L, Smítal J. Counterexamples to the open problem by Zhou and Feng on minimal center of attraction. Nonlinearity 2012;25:1443–9], shows that positive topological entropy supported by the center CzCz of attraction of a point z   is not related to the property that CzCz is the support of an invariant measure generated by z  . We also show that IR(f)IR(f) is invariant with respect to standard operations, like f(IR(f))=IR(f)f(IR(f))=IR(f), or IR(fm)=IR(f)IR(fm)=IR(f) for m∈Nm∈N.