Natural measures and statistical properties of non-statistical maps with multiple neutral fixed points

Abstract: In this article we show that a large class of infinite measure preserving dynamical systems that do not admit physical measures nevertheless exhibit strong statistical properties. In particular, we give sufficient conditions for existence of a distinguished natural measure $\nu$ such that the pushforwards of any absolutely continuous probability measure converge to $\nu$. Moreover, we obtain a distributional limit law for empirical measures. We also extend existing results on the characterisation of the set of almost sure limit points for empirical measures. Our results apply to various intermittent maps with multiple neutral fixed points preserving an infinite $\sigma$-finite absolutely continuous measure.