# Spectra of stationary processes on Z

## TL;DR

This paper explores a spectral property of stationary processes on Z, showing that a spectral gap implies the process is periodic, and discusses extensions and related open questions.

## Contribution

It establishes a novel link between spectral gaps and periodicity in finitely valued stationary processes on Z, extending previous results.

## Key findings

- Spectral gap implies process periodicity.
- Extensions of the main spectral property.
- Raises open questions for further research.

## Abstract

We will discuss a somewhat striking spectral property of finitely valued stationary processes on Z that says that if the spectral measure of the process has a gap then the process is periodic. We will give some extensions of this result and raise several related questions.