Explicit Correlation Bounds for Expanding Circle Maps Using the Coupling Method

TL;DR

This paper applies the modern coupling method to smooth expanding circle maps, deriving explicit bounds for key dynamical properties like ergodic measures and mixing rates, providing a clear introduction to the technique.

Contribution

It introduces the coupling method to derive explicit bounds for fundamental properties of expanding circle maps, a novel application in this context.

Findings

Existence and uniqueness of absolutely continuous ergodic measure

Explicit bounds on exponential mixing rate

Application of coupling method to dynamical systems

Abstract

In this paper, several fundamental facts, especially the existence and uniqueness of an absolutely continuous ergodic measure with an exponential mixing rate, are derived for smooth expanding circle maps. Although the results are classical, the coupling method is relatively modern and employed here for the first time to yield explicit bounds in terms of system constants. The work constitutes a part of the author's Bachelor's thesis. The author hopes that the manuscript can serve as a useful introduction to the flexible coupling method in the theory of dynamical systems.