András Némedy Varga (BME MI)
  High dimensional generalization of standard pairs and the coupling
  technique.
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Abstract: For uniformly hyperbolic dynamical systems with singularities one
particular technique to prove exponential decay of correlations is the
coupling of standard pairs. It was developed by Chernov and Dolgopyat for
systems with two dimensional phase spaces. The key idea is the following.
Consider any two standard pairs, which are just unstable curves with some
measures on them that have sufficiently regular densities. Iterating them
forward by the dynamics at certain times some parts of their images will be
very close to each other, so that they will be connected by stable manifolds
and hence the distance between them converges to zero exponentially fast. At
these times - due to the previous reason - the measures they carry may be
coupled along stable manifolds. If each time a fix amount of the measures
can be coupled and the measure of those points, who are not coupled at time
n is exponentially small in n, then the system enjoys exponential decay of
correlations and also some other statistical properties (e.g. limit
theorems) hold. In this talk I would like to present the (almost complete)
high dimensional generalization of this method, enlightening the
difficulties arising from the d > 2 setup.