On the topological entropy of (a,b)-continued fraction transformations

TL;DR

This paper investigates the topological entropy of (a,b)-continued fraction transformations, showing it remains constant on certain parameter regions and providing evidence of variability across the entire parameter space.

Contribution

It proves the constancy of topological entropy on a specific square in parameter space and offers experimental evidence of its flexibility elsewhere.

Findings

Topological entropy is constant on a square in parameter space.

Experimental evidence suggests entropy varies across the entire parameter space.

Maps conjugated to constant slope maps are used in the proof.

Abstract

We study the topological entropy of a two-parameter family of maps related to (a,b)-continued fraction algorithms and prove that it is constant on a square within the parameter space (two vertices of this square correspond to well-studied continued fraction algorithms). The proof uses conjugation to maps of constant slope. We also present experimental evidence that the topological entropy is flexible (i.e., takes any value in a range) on the whole parameter space.