Global-local mixing for the Boole map

TL;DR

This paper proves global-local mixing for the Boole map, demonstrating how the system's evolution leads to decorrelation between global and local observables, and convergence of measures to an average over the space.

Contribution

It establishes global-local mixing for the Boole map, a key example of a non-uniformly expanding map with neutral fixed points, advancing understanding of infinite-volume mixing.

Findings

Proves global-local mixing for the Boole map.

Shows convergence of measures to an averaging functional.

Demonstrates decorrelation of global and local observables.

Abstract

In the context of 'infinite-volume mixing' we prove global-local mixing for the Boole map, a.k.a. Boole transformation, which is the prototype of a non-uniformly expanding map with two neutral fixed points. Global-local mixing amounts to the decorrelation of all pairs of global and local observables. In terms of the equilibrium properties of the system it means that the evolution of every absolutely continuous probability measure converges, in a certain precise sense, to an averaging functional over the entire space.