2011/05/30 by Svetlana Katok, Ilie Ugarcovici, Katok, Svetlana +1 · 1 citation
Mathematics · #37B40 #37D40 #Attractor #Bernoulli's principle #Combinatorics #Countable set #Discrete mathematics #Dynamical Systems (math.DS) #Entropy (arrow of time) #FOS: Mathematics #Geodesic #Invariant measure #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #math.DS #msc:37B40 #msc:37D40
paper · pdf · doi:10.48550/arxiv.1105.6133
published in arXiv (Cornell University) (Cornell University) · accepted for publication, Ergodic Theory and Dynamical Systems
arxiv created 2011/05/30 · openalex publication_date 2011/05/30 · arxiv updated 2011/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
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.