### Pointwise ergodic averages along sequences of slow growth

Ergodic Theory and Dynamical Systems Seminar

7th November 2024, 2:00 pm – 3:00 pm

Fry Building, 2.04

Following the Birkhoff Ergodic Theorem, it is natural to consider whether convergence still holds along subsequences of the integers. Bergelson and Richter showed that for uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$ counted with multiplicities. However, removing the assumption of unique ergodicity, a pointwise ergodic theorem does not hold along $\Omega(n)$. In this talk, we will classify the strength of this non-convergence by considering weaker notions of averaging. In particular, we will show that double logarithmic averaging recovers pointwise convergence. Additionally, we will introduce a criterion for identifying other slow growing sequences satisfying a similar non-convergence property (based on joint work with S. Mondal).

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