Chain Mixing and Chain Recurrent Iterated Function Systems

TL;DR

This paper explores the properties of chain mixing, chain transitivity, and chain recurrence in iterated function systems, proving their equivalence using topological conjugacy and adding machine maps, and comparing these notions with discrete dynamical systems.

Contribution

It introduces the concepts of chain mixing and chain transitive iterated function systems and proves their equivalence, advancing the understanding of their ergodic properties.

Findings

Chain mixing, chain transitive, and chain recurrence are equivalent in iterated function systems.

Examples are provided to compare these notions with discrete dynamical systems.

The paper uses adding machine maps and topological conjugacy in the analysis.

Abstract

This paper considers the egodicity properties in iterated function systems. First, we will introduce chain mixing and chain transitive iterated function systems then some results and examples are presented to compare with these notions in discrete dynamical systems. As our main result, using adding machine maps and topological conjugacy we show that chain mixing, chain transitive and chain recurrence properties in iterated function systems are equivalent.