Dynamical Systems Seminar

When confronted with a smooth dynamical system that appears to possess some sort of non-uniform hyperbolicity, it is useful to find an invariant measure that controls the asymptotic properties of points chosen at random with respect to the natural volume on the phase space. Such SRB measures have been constructed for systems where it is possible to relate the dynamics to a symbolic system via a Markov partition or Young tower, and also for certain systems with a dominated splitting. We present a new approach that does not require any Markov structure or uniform geometric structure. The key is a notion of "effective hyperbolicity", which can be used to prove a non-uniform version of the Hadamard-Perron theorem on stable and unstable manifolds. This is joint work with Dmitry Dolgopyat and Yakov Pesin.