Journal of Modern Dynamics (JMD)

Structure of attractors for $(a,b)$-continued fraction transformations

Pages: 637 - 691, Issue 4, October 2010      doi:10.3934/jmd.2010.4.637

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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 related $(a,b)$-continued fractions suggested for consideration by Don Zagier. We prove that 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 Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed'' from the data $(a,b)$, and that 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 show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.

Keywords:  Continued fractions, attractor, natural extension, invariant measure.
Mathematics Subject Classification:  37E05, 11A55, 11K50.

Received: March 2010;      Revised: September 2010;      Available Online: January 2011.