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4:00 pm Thursday, January 10, 2013 Colloquium: Quantitative shrinking targets for IETs and rotationsby Jon Chaika (University of Chicago) in HB 227- In this talk we present some quantitative shrinking target results. Consider $T:[0,1] \to [0,1]$. One can ask how quickly under T a typical point $x$ approaches a typical point $y$. In particular given $\{a_i\}_{i=1}^{\infty}$ is $T^ix \in B(y,a_i)$ infinitely often? A finer question of whether $T^ix \in B(y,a_i)$ as often as one would expect will be discussed. That is, does $$ \underset{N \to \infty}{\lim}\frac{\underset{n=1}{\overset{N}{\sum}} \chi_{B(y,a_n)(T^nx)}}{\underset{n=1}{\overset{N}{\sum}} 2a_n}=1 $$ for almost every $x$. These results also apply to a billiard in the typical direction of any rational polygons. This is joint work with David Constantine.
Submitted by michael@rice.edu |