Distributional convergence of empirical measures and strong natural measures for non-statistical systems
Abstract: Maps with several equally and sufficiently sticky neutral fixed points can present non-statistical behaviour in the sense that the sequence of empirical measures does not converge for Lebesgue almost every initial condition. Such systems do not admit SRB measures or physical measures. We will show that nevertheless one can sometimes give a precise description of the long-term behaviour of the empirical measures in these situations. For example, one can determine the almost sure limit points of the empirical measures and prove that the empirical measures do in fact converge in distribution. Moreover, one can show the existence of a strong natural measure: a distinguished measure v so that the push forwards of any absolutely continuous measure converge to v. We will also discuss how there are some invariant measures whose basin of attractions are of full Hausdorff dimension, despite being of zero Lebesgue measure. These are joints works with Ian Melbourne and Amin Talebi, and with Katrin Gelfert.