TITLE: Uniquely ergodicity and polynomial ergodic average
QUESTION [8 upvotes]: Let $(X,T)$ be a uniquely ergodic system (here X is compact, T is a continuous map form $X$ to itself), so for any continuous function $f:X\rightarrow\mathbb{R}$ we have for any $x\in X$, the ergodic average $$\frac{1}{n}\sum_{i=0}^{n-1}f(T^ix)$$ is convergent pointwise (in fact it is uniformly convergent). Now I am interested in what happen to the polynomial ergodic average, that is, $$\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i^2}x).$$ Is this average able to converge? We know that this average is converge almost everywhere for $f\in L^p(\mu)$ with $p>1$, but not for $L^1$, from the work of Bourgain, Zoltán Buczolich , R. Daniel Mauldin and etc.. Now with $f\in C(X)$ can we ask for pointwise convergence, or there is an counterexample? If this is false, can we add some natural conditions on the system to make this true (natural here I mean that there can be a big class of these systems including some common systems not trivial ones)? Thanks for any idea, comment or reference.

REPLY [5 votes]: Theorems 1.2 and 1.3 of R. Pavlov's article Some counterexamples in topological dynamics provide strong counterexamples for the convergence of $\frac{1}{N}\sum_{n=1}^{N}f(T^{n^2}x)$; in particular there are totally minimal, totally uniquely ergodic, and topologically mixing systems on connected spaces $X$ where convergence fails.