Equidistribution of orbits at polynomial times in rigid dynamical systems
Kosma Kasprzak
TL;DR
This work studies equidistribution of orbits sampled at polynomial times in uniquely ergodic and weakly mixing dynamical systems. It introduces quantitative rigidity conditions that connect polynomial-exponent orbit behavior to residue-class distributions, enabling new equidistribution results, notably for square orbits $T^{n^2}$ and, under weaker rigidity, for $T^{n^C}$. The authors prove a strong-rigidity theorem yielding uniform equidistribution of $T^{n^2}$-orbits and construct examples (skew products and special flows) that satisfy the rigidity, including weakly mixing cases. They further develop a weak-rigidity framework, showing subsequence equidistribution via approximations by scaled characters and Pow$_N(i)$, along with counterexamples illustrating the method’s limitations and the necessity of certain boundedness conditions. The results advance understanding of polynomial-time equidistribution beyond nilsystems and offer a toolkit of harmonic-analytic and number-theoretic methods applicable to rigid dynamical systems with weak mixing.
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 $(ω(q_n))$ is bounded.