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Applications of (a,b)-continued fraction transformations

Published online by Cambridge University Press:  19 September 2011

SVETLANA KATOK  and
ILIE UGARCOVICI
SVETLANA KATOK
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email: katok_s@math.psu.edu)
ILIE UGARCOVICI
Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA (email: iugarcov@depaul.edu)
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Abstract

We describe a general method of arithmetic coding of geodesics on the modular surface based on the study of one-dimensional Gauss-like maps associated to a two-parameter family of continued fractions introduced in [Katok and Ugarcovici. Structure of attractors for (a,b)-continued fraction transformations. J. Modern Dynamics4 (2010), 637–691]. 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 739 - 761
Copyright
Copyright © Cambridge University Press 2011

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