Quenched limit theorems for random U(1) extensions of expanding maps

TL;DR

This paper extends spectral methods to establish quenched limit theorems, including CLT and large deviations, for random U(1) extensions of expanding maps on the torus, advancing understanding of their statistical properties.

Contribution

It introduces a spectral approach for quenched limit theorems in partially hyperbolic dynamics, building on previous spectral results for Lyapunov spectra.

Findings

Proved quenched central limit theorems for the systems.

Established large deviations principles.

Derived local central limit theorems.

Abstract

The Lyapunov spectra of random U(1) extensions of expanding maps on the torus were investigated in our previous work [NW2015]. Using the result, we extend the recent spectral approach for quenched limit theorems for expanding maps [DFGV2018] and hyperbolic maps [DFGV2019] to our partially hyperbolic dynamics. Quenched central limit theorems, large deviations principles and local central limit theorems for random U(1) extensions of expanding maps on the torus are proved via corresponding theorems for abstract random dynamical systems.