Distribution of polynomial orbits in Toeplitz systems

TL;DR

This paper investigates the behavior of polynomial orbits in Toeplitz systems, revealing conditions for convergence, divergence, and equidistribution, and highlighting differences between regular and strictly ergodic systems.

Contribution

It demonstrates that polynomial averages can diverge or converge differently within Toeplitz systems and clarifies the relationship between orbit density and equidistribution.

Findings

Averages along different polynomials can exhibit contrasting convergence behaviors.

Density of polynomial orbits implies equidistribution in regular Toeplitz systems.

In strictly ergodic systems, density does not necessarily imply equidistribution.

Abstract

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study density of the polynomial orbits in Toeplitz systems -- we show that it implies equidistribution of the polynomial orbits in the class of regular Toeplitz systems, but not in the class of strictly ergodic ones.