On Focusing Entropy at a Point

Abstract

In the paper we consider points focusing entropy and such that this fact is influenced exclusively by the behaviour of the function around these points (i.e., it is independent from the form of the function at any distance from these points). Thus the notion of an $\mathcal{F}$-focal entropy point has been introduced. We prove that each edge periodic tree function and each continuous function mapping the unit interval into itself have such points. Moreover, we discuss the possibility of improving functions defined on some topological manifolds so that any fixed point of the function becomes its focal entropy point.