The entropy of alpha-continued fractions: numerical results

TL;DR

This paper numerically investigates the metric entropy of alpha-continued fractions, revealing regularity on matching intervals, a hierarchical structure of these intervals, and complex behavior outside them.

Contribution

It provides a detailed numerical analysis of entropy behavior, characterizes the matching set M, and explores the regularity and irregularity of entropy across parameter space.

Findings

Matching intervals form a hierarchical structure.

Entropy is smooth on matching intervals.

Outside M, entropy can be non-monotone.

Abstract

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.