I will present recent joint work with I. Melbourne and A. Talebi in which we consider a class of dynamical systems $f$ which preserve an infinite absolutely continuous ergodic measure $\mu$, but that are non-statistical (or historic) and so do not admit any physical measures (for example, our results apply to certain intermittent interval maps with two equally sticky fixed points). Using results from operator renewal theory we prove, despite the lack of physical measures, that there exits an invariant probability measure $\nu$ such that the push forward $f_*^n \lambda$ of any absolutely continuous probability measure $\lambda$ converges to $\nu$. Moreover, we give a precise description of the asymptotic distribution of the empirical measures $e_n$ in the space of probability measures. Namely, we show that despite the fact that the empirical measures $e_n$ do not converge almost surely they do converge in distribution.

This presentation will be divided into three 1-1.5 hour talks. In the first talk I will introduce our results and give some background from infinite ergodic theory. In the second talk we will give a very brief introduction to operator renewal theory and discuss a key result due to S. Gouëzel (2010) and I. Melbourne, D. Terhesiu (2011). Finally, in the third talk we will give a brief overview of how the result of Gouëzel, Melbourne, Terhesiu can be used give a proof of the results presented in the first talk.