Infinite mixing for one-dimensional maps with an indifferent fixed point

TL;DR

This paper investigates infinite-volume mixing properties of specific one-dimensional intermittent maps with indifferent fixed points, establishing global-local mixing and deriving related limit theorems.

Contribution

It introduces new results on global-local mixing for classes of intermittent maps with indifferent fixed points, including the Farey and Pomeau-Manneville maps.

Findings

Proved global-local mixing for Farey and Pomeau-Manneville maps.

Established limit theorems based on mixing properties.

Analyzed infinite-volume mixing in maps with indifferent fixed points.

Abstract

We study the properties of `infinite-volume mixing' for two classes of intermittent maps: expanding maps $[0,1] \longrightarrow [0,1]$ with an indifferent fixed point at 0 preserving an infinite, absolutely continuous measure, and expanding maps $\mathbb{R}^+ \longrightarrow \mathbb{R}^+$ with an indifferent fixed point at $+\infty$ preserving the Lebesgue measure. All maps have full branches. While certain properties are easily adjudicated, the so-called global-local mixing, namely the decorrelation of a global and a local observable, is harder to prove. We do this for two subclasses of systems. The first subclass includes, among others, the Farey map. The second class includes the standard Pomeau-Manneville map $x \mapsto x+x^2$ mod 1. Morevoer, we use global-local mixing to prove certain limit theorems for our intermittent maps.