Mean dimension of general iterated function systems

TL;DR

This paper introduces and studies Mean Dimension and Metric Mean Dimension for generalized iterated function systems, establishing their properties and relations, and linking the Gluing Orbit Property to positive topological entropy.

Contribution

It defines new invariants for generalized IFS, proves bounds and properties, and connects the Gluing Orbit Property to entropy under certain conditions.

Findings

Mean Dimension is bounded by Metric Mean Dimension

Systems with Small Boundary Property have zero Mean Dimension

Gluing Orbit Property implies positive topological entropy

Abstract

In this paper, we introduce and investigate the notions of Mean Dimension and Metric Mean Dimension for generalized iterated function systems (IFS). We establish basic properties of these invariants and prove that Mean Dimension is always bounded above by the lower Metric Mean Dimension and the upper Metric Mean Dimension in this setting. We further show that generalized iterated function systems with the Small Boundary Property have zero Mean Dimension. Finally, we introduce a Gluing Orbit Property for generalized iterated function systems and prove that, under suitable transitivity and non-rigidity assumptions, it guarantees positive topological entropy.