Dynamical Systems Seminar

Title: Grid method for divergence of averages
Speaker: Sovanlal Mondal
Speaker Info: University of Memphis
Brief Description:
Special Note:

Let $(T^t)_{t\geq 0}$ be a continuous, measure preserving and aperiodic flow on a probability space $(X,\Sigma, \mu)$, and $\alpha$ be a non integer rational number. For a function $f\in L^1$, we consider the ergodic averages $\frac{1}{N}\sum_{n\leq N}f(T^{n^\alpha}x)$ which is obtained by sampling the flow $(T^t)$ along $(n^\alpha)$-th times. In this talk, we prove that for every $\epsilon>0$, there is a set $E\in \Sigma$ with $\mu(E)<\epsilon$ such that the ergodic averages satisfies $\displaystyle \limsup_{N\to\infty}\frac{1}{N}\sum_{n\leq N}1_E(T^{n^\alpha}x)=1$ and $\displaystyle \liminf_{N\to\infty}\frac{1}{N}\sum_{n\leq N}1_E(T^{n^\alpha}x)=0.$ This is the worst possible divergence for indicators. This result is an improvement of a result of V. Bergelson, M. Boshernitzan and J. Bourgain which says that the averages along $(n^\alpha)$ diverge a.e. Note that the averages do converge in $L^2$ norm. The method we use is quite general and can be used to settle other open problems or improve other results. In the talk we mention some quite old open problems.
Date: Tuesday, May 02, 2023
Time: 4:00pm
Where: Lunt 104
Contact Person: Prof. Bryna Kra
Contact email: kra@math.northwestern.edu
Contact Phone: 847-491-5567
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