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

Abstract

We study the properties of `infinite-volume mixing' for two classes of intermittent maps: expanding maps $[0,1]⟶[0,1]$ with an indifferent fixed point at 0 preserving an infinite, absolutely continuous measure, and expanding maps ${\mathbb{R}}^{+}⟶{\mathbb{R}}^{+}$ 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.

arXiv commentFinal version to be published in Nonlinearity. 39 pages, 2 figures

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Cited by in corpus (4)

• Pointwise convergence of Birkhoff averages for global observables
• Pomeau-Manneville maps are global-local mixing
• Global-local mixing for the Boole map
• Quantitative global-local mixing for accessible skew products