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

TL;DR

This paper investigates infinite measure preserving dynamical systems with multiple neutral fixed points, establishing conditions for natural measures and distributional limit laws despite the absence of physical measures.

Contribution

It provides new criteria for the existence of natural measures and extends results on empirical measure limit points in systems with neutral fixed points.

Findings

Existence of a distinguished natural measure $ u$ under certain conditions

Distributional limit law for empirical measures in these systems

Characterization of almost sure limit points for empirical measures

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.