Random Lochs' Theorem

TL;DR

This paper extends Lochs' Theorem to random dynamical systems, providing a probabilistic framework for understanding the relationship between different transformations and establishing a Central Limit Theorem.

Contribution

It introduces a version of Lochs' Theorem for random systems and develops a random entropy formula, advancing the understanding of cylinder set sizes in stochastic dynamics.

Findings

Proves Lochs' Theorem for random dynamical systems.

Establishes a Central Limit Theorem in this context.

Provides a random Rokhlin's entropy formula.

Abstract

In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number $x$ that can be determined from just knowing its first $n$ decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs' Theorem for random dynamical systems as well as a corresponding Central Limit Theorem. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin's formula for entropy.