Description:

Title: Equidistribution along square orbits in rigid dynamical systems

Abstract: One of the ways to generalize the classical concept of
ergodic averages is as follows: instead of considering average values
of some function $f$ along the whole orbit $(T^n(x))$, consider them
only along a subsequence $(T^{a_n}(x))$. We focus on the case when
$(a_n)=(P(n))$ for some polynomial $P$, and let $(X, T)$ be a uniquely
ergodic topological dynamical system. We are then interested in the
property that such ergodic averages along $(a_n)$ converge for every
$x\in X$ to the integral of $f$. This is a very delicate property, and
not many examples of such systems are known.
We will see a new method for establishing this kind of property for
$(a_n)=(n^2)$ by assuming a strong rigidity condition on $X$, in
particular obtaining weakly mixing examples. Here by rigidity we mean
the existence of a sequence $(T^{q_n})$ of iterates of $T$ converging
to the identity map on $X$ — we will in in fact require a uniform,
quantitative form of this convergence.
The method relies on input from number theory about the distribution
of square residues in arithmetic progressions, and interestingly it
does not seem to yield results for polynomials of larger degree. We
will discuss the reason for this, as well as examples of systems
satisfying the required rigidity condition.