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4:00 pm Wednesday, September 2, 2009 Geometry-Analysis Seminar: Shrinking Targets for Interval Exchange Transformations and Sequencesby
Jon Chaika (Rice University) in HB 227- Let $T\colon [0,1) \to [0,1)$ be ergodic. It is easy to see that for any $\epsilon>0$ and almost every $x$ the sets $\LS B(T^ix,\epsilon)$ and $\LS T^{-i}B(x,\epsilon)$ have full measure. One can ask for quantitative analogues of this. That is, pick a sequence $\epsilon_1,\epsilon_2,...$ does $\LS B(T^ix,\epsilon_i)$ or $\LS T^{-1}B(x,\epsilon_i)$ have full measure for almost every (or every) $x$? The Borel-Cantelli Theorem presents an obvious requirement, $\underset{n=1}{\overset{\infty}{\sum}}\epsilon_n=\infty$. It is also natural to restrict our attention to non-increasing sequences. Results will be presented for IETs and general sequences. Some of this is joint work with Michael Boshernitzan.
Submitted by damanik@rice.edu |