Tuning and plateaux for the entropy of $\alpha$-continued fractions

TL;DR

This paper investigates the fractal structure and self-similarity of the entropy function for alpha-continued fractions, characterizing plateaux and monotonicity regions to deepen understanding of its complex behavior.

Contribution

It provides a detailed analysis of the fractal structure of the entropy function and fully characterizes its plateaux and monotonicity properties.

Findings

Identification of the fractal structure of the exceptional set

Complete classification of plateaux in the entropy graph

Description of the local monotonic behavior of the entropy function

Abstract

The entropy $h(T_\alpha)$ of $\alpha$-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set $\EE$. We will exploit the explicit description of the fractal structure of $\EE$ to investigate the self-similarities displayed by the graph of the function $\alpha \mapsto h(T_\alpha)$. Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.