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

Svetlana Katok, Ilie Ugarcovici

Research output: Contribution to journalArticle

12 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)637-691
Number of pages55
JournalJournal of Modern Dynamics
Volume4
Issue number4
DOIs
StatePublished - Oct 1 2010

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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