Kosma Kasprzak (Jagiellonian University)
Abstract: One of the main trends in sparse ergodic theory is examining ergodic averages on a dynamical system $(X, T)$ along sparse orbits of the form $(T^{P(n)}(x))$, for polynomials $P$. We will focus on the cases where the sequence of such averages converges for every starting point, and in particular when each polynomial orbit is equidistributed in $X$. These are very delicate properties, and not many such systems are known — we lack examples with mixing properties or ones exhibiting different behavior for different choices of $P$.
I will introduce the known approaches and discuss the above limitations, before focusing on an original result stating that the desired equidistribution holds under a rigidity assumption. Here by rigidity we mean the existence of a sequence $(T^{q_n})$ of iterates of $T$ converging to the identity map on $X$ — in our case we need strong quantitative bounds on the rate of convergence, which however still allow for weakly mixing examples.