course: Minicourse 3 Effective Mixing, Effective Equidistribution and Renormalization
speaker: Livio Flaminio (Université Lille 1)abstract: It is well known that for Anosov systems the effective equidistribution of stable foliation is equivalent to some effective mixing property (decay of correlation) for the Anosov system. In the first part of the course we shall review the case of classical horocycle flows. For these flows effective equidistribution can be proved by methods of harmonics analysis together with the renormalization methods. For the classical horocycle flow, the natural “renormalization flow” is given by the geodesic flow. Harmonic analysis is used to obtain tame estimates for the solutions of the cohomological equation for the horocycle flow and renormalization to estimate the projections of the currents associated to orbit intervals onto the space of flow-invariant currents. In the second part of the course we shall shows how to apply these ideas to the case of the stable foliation of partially hyperbolic automorphisms of some nilmanifolds. As result we obtain effective mixing of these partially hyperbolic automorphisms. Finally we shall consider the generic case of families of flows on Heisenberg nilmanifolds that are not immediately renormalizable by a diffeomorphism of the ambient manifold. Such flows form a bundle on which a recurrent renormalization dynamics can be defined. Using the bundle renormalization dynamics we can recover effective equidistribution for the “generic” Heisenberg nilflow. This approach was inspired by the renormalization of translation flows on compact surfaces, which is given by the Teichmueller flow on the moduli space of holographic differentials. If time permits we shall show how one can handle the case of flows for which no recurrent renormalization dynamics exists.
timetable:
Tue 28 Jun, 9:30 - 10:30, Aula Dini
Tue 28 Jun, 11:00 - 12:00, Aula Dini
Wed 29 Jun, 9:00 - 10:00, Aula Dini
Wed 29 Jun, 14:00 - 15:00, Aula Dini
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