Colloque des sciences mathématiques du Québec

25 septembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

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

Colloque par Svetlana Katok

I will discuss one-dimensional maps related to a family of (a,b)-continued fractions, suggested for consideration by Don Zagier, and give a sufficient condition for validity of the Reduction theory conjecture that states that the associated natural extension maps have attractors with finite rectangular structure where every point of the plane is mapped after finitely many iterations. I will show how the structure of these attractors can be computed from the data $(a,b)$, and give a dynamical interpretation of the ``reduction theory" that underlines these constructions. The set of parameter pairs $(a,b)$ for which the conjecture is not valid is also well-understood; in particular, the points for which the attractors do not have finite rectangular structure is a non-empty nowhere dense subset of the boundary $b=a+1$ of the set of parameters . If time permits, I will also explain how these continued fractions can be used for coding of geodesics on the modular surface. This is a joint work with Ilie Ugarcovici.
 

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UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420