UIUC Dept. of Mathematics Seminar Calendar

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                        30

Monday, September 17, 2012

Ergodic Theory
4:00 pm in 241 Altgeld Hall, Monday, September 17, 2012

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Andrew Parrish (UIUC)
Convergence of Sparse Subset Averages of $L^1$ Functions.
Abstract: The behavior of time averages when taken along subsets of the integers is a central question in subsequence ergodic theory. The existence of transference principles enables us to talk about the convergence of averaging operators in a universal sense; we say that a sequence $\{a_n\}$ is universally pointwise good for $L^1$, for example, if the sequence of averages \begin{equation*} \frac{1}{N} \sum_{n=0}^{N-1} f \circ \tau^{-a_n}(x) \end{equation*} converges a.e. for any $f\in L^1$ for every aperiodic measure preserving system $(X, \mathcal{B}, \mu, \tau)$. Only a few methods of constructing a sparse sequence that is universally pointwise $L^1$-good are known. We will discuss how one can construct families of sets in $\mathbb{Z}^d$ which are analogues of these sequences, as well as some challenges and advantages presented by these higher-dimensional averages. Joint work with P. LaVictoire (University of Wisconsin, Madison) and J. Rosenblatt (UIUC).