The entropy of alpha-continued fractions: analytical results

TL;DR

This paper investigates the ergodic properties of a family of interval maps related to generalized continued fractions, establishing the Hölder continuity of entropy and proving a central limit theorem for certain observables.

Contribution

It provides new analytical results on the entropy dependence and statistical behavior of a family of continued fraction maps, expanding understanding of their ergodic theory.

Findings

Entropy varies Hölder-continuously with the parameter alpha

A central limit theorem is established for a broad class of observables

Results apply to Birkhoff averages converging to the entropy

Abstract

We study the ergodic theory of a one-parameter family of interval maps T_alpha arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of T_alpha to be Hoelder-continuous in the parameter alpha. Moreover, we prove a central limit theorem for possibly unbounded observables whose bounded variation grows moderately. This class of functions is large enough to cover the case of Birkhoff averages converging to the entropy.