Year: 2011
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Katok, Svetlana, Ugarcovici, Ilie
Institutions: Pennsylvania State University, DePaul University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2951822196