|4:00 pm Monday, March 19, 2012|
Special Colloquium: Cocycles Over Rotations and Interval Exchange Transformations, Application to the Billiard in the Plane
by Jean-Pierre Conze (University of Rennes 1) in HB 227
Since the end of the seventies, the cocyles have played an important role in the study of dynamical systems with infinite measure.Host Department: Rice University-Mathematics
Formally, let (X, Î¼, Ï„ ) be a dynamical system, with Î¼ a probability measure on the space X invariant by the map Ï„, and let Î¦ be a measurable function from X to Rd . Generalizing the notion of random walk, a "cocycle" (Î¦n)nâ‰¥1 is defined by Î¦n(x) = ∑nâˆ’1k=0 Î¦(Ï„kx). The associated skew product is the map acting on X Ã— Rd by Ï„Î¦: (x,y)â†’(Ï„x,y+Î¦(x)).
The main problems are recurrence of the cocyle (Î¦n) and ergodicity of the map Ï„Î¦.
The representation of a dynamical system with infinite measure as a skew product is possible when there is a group (here Rd) acting on the phase space, commuting with the dynamics and with a quotient of finite measure. This situation happens for instance for the billiard with periodic obstacles.
A new interest emerged in the recent years via the study of ergodic properties of the billiard with periodically distributed obstacles in the plane, in particular for rectangular obstacles. The billiard map reduces to skew prod- ucts over interval exchange transformations. The cocycles correspond to displacement functions.
The aim of the talk with be to present results for cocycles over a rotation on the circle, a simple but already rich model. We will also address some results on the billiard in the plane, a domain very active with new and sometimes surprising results due to several authors.