Concentration inequalities for sequential dynamical systems of the unit interval

TL;DR

This paper establishes a concentration inequality for sequential dynamical systems on the unit interval with exponential memory loss, impacting the understanding of compositions of beta-transformations near a fixed parameter.

Contribution

It introduces a concentration inequality applicable to systems with exponential memory decay, including compositions of beta-transformations close to a specific parameter.

Findings

Proves a concentration inequality for systems with exponential BV norm loss

Applies results to compositions of beta-transformations near a fixed beta

Analyzes consequences of the concentration inequality in dynamical systems

Abstract

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BV norm, and we investigate several of its consequences. In particular, this covers compositions of $\beta$-transformations, with all $\beta$ lying in a neighborhood of a fixed $\beta_{\star} > 1$ and systems satisfying a covering type assumption.