Jon Chaika – Quantitative shrinking targets for IETs and rotations

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Date(s) - 24 juin 2011

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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 $\lim_{N \to \infty} \frac{\sum_{n=1}^N \chi_{B(y,a_n)}(T^nx)}{\sum^N_{n=1} 2a_n}=1$ for almost every $x$. We will present applications to billiards in rational polygons and a related result for Sturmian sequences. This is joint work with David Constantine.