Lai-Sang Young

Abstract:   About 10 years ago, I proposed a unified approach for obtaining statistical information on a class of chaotic dynamical systems. The idea is to exchange the inherent messiness of dynamical structures for the relative simplicity of certain countable state Markov chains. This can sometimes be done by constructing a ``Markov tower" over the dynamical system in question, but the construction is not without cost, and the comparison is not perfect: after all, dynamical systems are not Markov chains. I would like to report on some of the results that have been obtained since. Via examples, I will address the following two questions: (1) what kind of information does this method yield, and (2) for which types of systems has it been fruitful? In relation to (1) I will report on results on rates of correlation decay, large deviations, escape rates and surviving distributions for systems with leaks. As examples of (2) I will discuss billiards and strange attractors arising from the periodic forcing of oscillators.