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 $$\frac{1}{n}\sum_{i=0}^{n1}f(T^ix)$$ is convergent pointwise (in fact it is uniformly convergent). Now I am interested in what happen to the polynomial ergodic average, that is, $$\frac{1}{n}\sum_{i=0}^{n1}f(T^{i^2}x).$$ 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.
3 Answers
This is indeed true for some "nice systems", for example one can show this theorem (for say $L^{2}$functions) for Kronecker systems simply by vanderCorput trick.
In general, those averages converge everywhere for Nilmanifolds (Leibman, GreenTao) with full assorted measureclassification and orbit classification result. One cannot expect such a result to hold in general even if the system is uniquely ergodic (think for example about the example by Katznelson and Furstenberg for uniquely ergodic system with positive entropy).
There is active research in this problem for the case of unipotent dynamics. Venkatesh proved convergence of averages of the form $\frac{1}{N}\sum f(T^{i^{1.01}}.x)$ for uniform quotients of $PSL_{2}$. The exponent cannot be improved to $2$ even under Ramanujan. SarnakUbis managed to show a very weak result about the averages along the primes (which is also related to Bourgain's theorem that you've mentioned).
It's worth mentioning an old result due to Shah which shows convergence of those polynomial averages as long as you consider the continuoustime averages. Unfortunately, as this result relays on Ratner's theorem, it is unclear how to deduce the discrete analogue (it's actually an open conjecture of Margulis and Shah that those averages do converge everywhere), one of the main difficulties (in the nonuniform case) is that we don't even know nondivergence result.
There's a recent result which shows that for say oneparameter unipotent action on $G/\Gamma$ the Hausdorff dimension of the exceptional set for convergence of polynomial averages is small, but this is an almost everywhere theorem.
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.
This seems to be discussed at great length in the HostKra paper. They have many references.

1$\begingroup$ I think the things discuss in the HostKra paper are on the $L^2$ convergence of the multiple ergodic average. What I want to know is about pointwise convergence the ergodic average. I think they are not the same. $\endgroup$ Apr 15, 2015 at 2:18

$\begingroup$ @SimingTu They discuss all manner of convergence (at least via references, but also directly, I believe) $\endgroup$ Apr 15, 2015 at 2:20

2$\begingroup$ I believe the OP is asking about everywhere convergence. I don't see that addressed in HK. $\endgroup$ Apr 15, 2015 at 5:33