Ergodic Functions That are not Almost Periodic Plus $L^1-$Mean Zero

TL;DR

This paper investigates whether all ergodic functions can be decomposed into an almost periodic part plus an $L^1$-mean zero part, providing a framework that constructs many ergodic functions not fitting this decomposition.

Contribution

The paper introduces a framework demonstrating the existence of ergodic functions that are not decomposable into an almost periodic plus $L^1$-mean zero component, challenging previous assumptions.

Findings

Constructs infinitely many ergodic functions not decomposable as such.

Provides a theoretical framework for identifying non-decomposable ergodic functions.

Challenges the notion that all ergodic functions are sums of almost periodic and $L^1$-mean zero functions.

Abstract

Ergodic Functions are bounded uniformly continuous $(\text{BUC})$ functions that are typical realizations of continuous stationary ergodic process. A natural question is whether such functions are always the sum of an almost periodic with an $L^1-$mean zero $\text{BUC}$ function. The paper answers this question presenting a framework that can provide infinitely many ergodic functions that are not almost periodic plus $L^1-$ mean zero.