Equidistribution of orbits at polynomial times in rigid dynamical systems

TL;DR

This paper investigates the distribution of orbits at polynomial times in uniquely ergodic systems, establishing conditions under which such orbits are equidistributed or dense, including for weakly mixing systems.

Contribution

It introduces new rigidity conditions that guarantee equidistribution or density of polynomial orbits in uniquely ergodic and weakly mixing systems.

Findings

Square polynomial orbits are equidistributed under certain rigidity conditions.

Weakly mixing systems can satisfy these rigidity conditions, leading to equidistribution.

Weaker rigidity conditions imply orbit density in totally uniquely ergodic systems.

Abstract

We study distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems $(X, T)$. First, we prove that if there exists an increasing sequence $(q_n)$ for which the rigidity condition \[ \max_{t<q_{n+1}^{4/5}}\sup_{x\in X}d(x, T^{tq_n}x)=o(1) \] is satisfied, then all square orbits $(T^{n^2}x)$ are equidistributed (with respect to the only invariant measure). We show that this rigidity condition might hold for weakly mixing systems, and so as a consequence we obtain first examples of weakly mixing systems where such an equidistribution holds. We also show that for integers $C>1$ a much weaker rigidity condition \[ \max_{t<q_n^{C-1}}\sup\limits_{x\in X}d\left(x, T^{tq_n}x\right)=o(1) \] implies density of all orbits $(T^{n^C}x)$ in totally uniquely ergodic systems, as long as the sequence $(\omega(q_n))$ is bounded.