Monday 20 Nov 2017: Dynamics Seminar: Exponential equidistribution of standard pairs for piecewise expanding maps of metric spaces
Peyman Eslami - Warwick
For a large class of piecewise expanding maps of metric spaces we show the equidistribution of standard pairs at an exponential rate. As a corollary such systems have a unique absolutely continuous invariant measure with respect to which the system is mixing. We allow for unbounded, non-compact spaces, countably many branches and do not assume big images or the existence of a Markov structure. We show how to control the complexity growth of the dynamical partition of the map. Such control is necessary and crucial for systems that are not one-dimensional. Our method gives explicit estimates on the exponential rate of equidistribution. If there is time, I will also comment on the construction of Banach spaces (made out of standard pairs) on which the transfer operator is quasi-compact. If one is not concerned with explicit bounds on the constants involved in decay of correlations, this functional analytic point of view leads to establishing further statistical properties of the system in a standard manner.