EVENT DETAILS AND ABSTRACT

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

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.