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

Svetlana Katok; Ilie Ugarcovici

doi:10.3934/jmd.2010.4.637

Article Contents

2010, Volume 4, Issue 4: 637-691. Doi: 10.3934/jmd.2010.4.637

This issue Previous Article Ratner's property and mild mixing for special flows over two-dimensional rotations Next Article Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory

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

Svetlana Katok^1, and
Ilie Ugarcovici^2,

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
2.
Department of Mathematical Sciences, DePaul University, 2320 N. Kenmore Ave., Chicago, IL 60614-3504

Received: February 28, 2010

Revised: August 31, 2010

Published: December 2010

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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.

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Mathematics Subject Classification: 37E05, 11A55, 11K50.

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