The Entropy of an Overlapping Dynamical System

TL;DR

This paper investigates the entropy of a specific class of overlapping dynamical systems, establishing conditions for homeomorphisms between fractal attractors and deriving a formula for their topological entropy.

Contribution

It introduces a correspondence between overlapping functions and IFSs to determine when fractal transformations are homeomorphisms and provides a formula for their topological entropy.

Findings

Necessary and sufficient condition for homeomorphism between attractors

Formula for topological entropy of overlapping functions

Characterization of overlapping IFS dynamics

Abstract

The term "overlapping" refers to a certain fairly simple type of piecewise continuous function from the unit interval to itself and also to a fairly simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used (1) to find a necessary and sufficient condition for a fractal transformation from the attractor of one overlapping IFS to the attractor of another overlapping IFS to be a homeomorphism and (2) to find a formula for the topological entropy of the dynamical system associated with an overlapping function.