Reduction theory for Fuchsian groups and coding of geodesics.

Géométrie Dynamique

Salle Duhem M3
Svetlana Katok
Penn State University
Vendredi, 15 Septembre, 2017 - 10:00 - 11:00

I will discuss a method of coding of geodesics on quotients of the hyperbolic plane by Fuchsian groups using boundary maps and “reduction theory”.

For the modular surface these maps are related to a family of (a,b)-continued fractions, and for compact surfaces these are generalizations of the Bowen-Series map.

The boundary maps are given by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuity points the associated two-dimensional natural extension maps possess global attractors with finite rectangular structure to which (almost) every point is mapped after finitely many iterations. These two properties belong to the list of “good” reduction algorithms, equivalence or implications between which were suggested by Don Zagier.  I will also explain how  the geodesic flow can be represented as a special flow over a cross-section of “reduced” geodesics parametrized by the corresponding attractor.  The talk is based on joint works with Ilie Ugarcovici and Adam Zydney.