Pointwise Ergodic Averages along Sequences of Slow Growth
Vortrag von Dr. Katy Loyd
Sprecher eingeladen von: Prof. Dr. Corinna Ulcigrai
Datum: 16.10.24 Zeit: 13.30 - 14.30 Raum: Y27H28
Following Birkhoff's proof of the Pointwise Ergodic Theorem, it is natural to consider whether convergence still holds along various subsequences of the integers. In 2020, Bergelson and Richter showed that in uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$, with multiplicities. In this talk, we will see that by removing this assumption, a pointwise ergodic theorem does not hold along $\Omega(n)$. In fact, $\Omega(n)$ satisfies a notion of non-convergence called the strong sweeping out property. We then further classify the strength of this non-convergence behavior by considering weaker notions of averaging. Time permitting, we will introduce a more general criterion for identifying slow growing sequences with the strong sweeping out property (based on joint work with S. Mondal).