Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Theory of $(a,b)$-continued fraction transformations and applications

Pages: 20 - 33, January 2010      doi:10.3934/era.2010.17.20

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Svetlana Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)
Ilie Ugarcovici - Department of Mathematical Sciences, DePaul University, 2320 N. Kenmore Ave., Chicago, IL 60614-3504, United States (email)

Abstract: We study a two-parameter family of one-dimensional maps and the related $(a,b)$-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.

Keywords:  Continued fractions, attractors, modular surface, invariant measure.
Mathematics Subject Classification:  Primary 37D40, 37B40; Secondary 11A55, 20H05.

Received: November 2009;      Revised: February 2010;      Available Online: April 2010.