On $$\mathbb {P}$$ P -Weakly Hyperbolic Iterated Function Systems

Abstract

In this paper we will consider the concept of \(\mathbb {P}\)-weakly hyperbolic iterated function systems on compact metric spaces that generalizes the concept of weakly hyperbolic iterated function systems, as defined by Edalat (Inf Comput 124(2):182–197, 1996) and by Arbieto, Santiago and Junqueira (Bull Braz Math Soc New Ser 2016) for a more general setting where the parameter space is a compact metric space. We prove the existence and uniqueness of the invariant measure of a \(\mathbb {P}\)-weakly hyperbolic IFS. Furthermore, we prove an ergodic theorem for \(\mathbb {P}\)-weakly hyperbolic IFS with compact parameter space.