Title: Correlation decay in multidimensional dispersing billiards
Abstract:
I describe a result proving that in a class of multi-dimensional
dispersing billiard maps the time correlations decay exponentially for
Holder-continuous observables. The result is joint with Peter Balint.
The class studied consists of the "simplest possible"
multi-dimensional dispersing billiards - namely those which have finite
horizon and no corner points, and in addition, a "subexponential
complexity condition" holds for the set of singularities. This is one of
the very few high (>2) dimensional systems of physical origin where such
strong correlation decay is rigorously proven.
In the first half of the talk I decsribe the setting, the main
features of the systems discussed, some of the historical background,
the main result and some of its consequences. I also comment on the
strange "subexponential complexity condition" that we need.
In the second half I talk about the method of the proof - the
well-known Young tower construction, and especially one key element of
the construction, which is a "growth lemma" describing the evolution of
unstable manifolds under the dynamics.