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Examples of dynamical systems with innite invariant measure

Summary: Examples of dynamical systems
with innite invariant measure
Jean-Pierre Conze
(University of Rennes 1)
Abstract : We discuss two models of dynamical systems with an innite invariant mea-
In a rst part, we consider the non-dispersive billiard ow given by the movement of a ball
in the plane with identical rectangular obstacles which are Z2
-periodically distributed.
The behavior of this ow is quite dierent from the billiard ow with dispersive barriers.
Here the phase space can be decomposed into invariant subsets A corresponding to the
4 directions that can take the ow from one initial direction . For this model of billiards
in the plane, the problems are : recurrence (does the ball go back to any neighborhood of
its initial point ?) and ergodicity on the sets A for the innite measure associated with
the Lebesgue measure in the plane.
We will show that, for a very particular choice of the direction ( = /4), the billard is
recurrent (this was shown in 1980 by J. Hardy and J. Weber) and ergodic if the ratio of
the length and the width of the rectangle is irrational.
In a second part (work in collaboration with Nicolas Chevallier), we will discuss the recur-
rence for extensions of bi-dimensional rotations and give examples and counter-examples


Source: Assani, Idris - Department of Mathematics, University of North Carolina at Chapel Hill


Collections: Mathematics