Quenched CLT for random toral automorphism

TL;DR

This paper proves a quenched Central Limit Theorem for smooth observables in random hyperbolic toral automorphisms, showing almost sure variance equivalence with the annealed case using transfer operator techniques.

Contribution

It introduces a novel quenched CLT for random hyperbolic maps on the torus and demonstrates variance equivalence with the annealed CLT.

Findings

Quenched CLT established for random toral automorphisms.

Variance of quenched system matches annealed system almost surely.

Transfer operator method on anisotropic Banach space is effective.

Abstract

We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.