Ergodic automorphisms with simple spectrum characterized by fast correlation decay

TL;DR

This paper constructs ergodic automorphisms with simple spectrum that exhibit rapid correlation decay at a rate close to $|k|^{-1/2}$ for a dense set of functions, advancing understanding of spectral and mixing properties.

Contribution

It demonstrates the existence of measure-preserving ergodic automorphisms with simple spectrum and specified fast correlation decay rates, a novel result in ergodic theory.

Findings

Existence of automorphisms with simple spectrum and correlation decay rate $O(|k|^{-1/2+ ext{epsilon}})$

Correlation decay holds for a dense family of functions

Advances understanding of spectral properties and mixing rates in ergodic systems

Abstract

The existence of measure preserving invertible transformations $T$ with simple spectrum is established possessing the following rate of correlation decay $(f(T^k x), f(x)) = O(|k|^{-1/2+{\epsilon}})$ for a dense family of functions $f$ and any ${\epsilon > 0}$.