Applications of (a,b)-continued fraction transformations

TL;DR

This paper introduces a general method for arithmetic coding of geodesics on the modular surface using (a,b)-continued fraction transformations, analyzing their structure, coding sequences, and measure-theoretic properties.

Contribution

It develops a unified approach to coding geodesics with (a,b)-continued fractions, including dual expansions, and characterizes the natural extension maps as Bernoulli shifts.

Findings

The attractors have a finite rectangular structure.

Coding sequences form a countable sofic shift in special cases.

Natural extension maps are Bernoulli shifts with computable entropy.

Abstract

We describe a general method of arithmetic coding of geodesics on the modular surface based on a two parameter family of continued fraction transformations studied previously by the authors. The finite rectangular structure of the attractors of the natural extension maps and the corresponding "reduction theory" play an essential role. In special cases, when an (a,b)-expansion admits a so-called "dual", the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. We also prove that the natural extension maps are Bernoulli shifts and compute the density of the absolutely continuous invariant measure and the measure-theoretic entropy of the one-dimensional map.