**Title:** Billiards in polygons

**Speaker:** Jon Chaika

**Speaker Info:** University of Utah

**Brief Description:**

**Special Note**:

**Abstract:**

Consider a point mass traveling in a polygon. It travels in a straight line, with constant speed, until it hits a side, at which point it obeys the rules of elastic collision. What can we say about this? When all the angles of the polygon are rational multiples of pi, it is directly related to translation surfaces and we can say a lot about it. In the case when at least one of the angles is irrational, it is much less understood, though from approximating with the rational case we know a couple of things. Kerckhoff, Masur and Smillie proved that there exists a billiard in an irrational polygon where the billiard flow is 'ergodic' with respect to the natural measure. This means that the amount of time the typical trajectory spends in a given box in the table (or even a cube in the three dimensional unit tangent bundle) is proportional to its area (or volume). The new result presented in this talk is a strengthening of this: there is an irrational polygon where the billiard flow is 'weakly mixing.' That is, the different trajectories are usually uncorrelated and it is joint work with Giovanni Forni.This talk will connect billiards in rational polygons to translation surfaces. It will state the results of Kerckhoff, Masur, Smillie and of Forni and myself for translation surfaces, which imply the results for billiards in certain polygons. It will describe the approximation argument, using the Baire Category theorem, by which the results for billiards are obtained from the results for translation surfaces. Open questions will be presented and no previous knowledge of billiards, ergodic theory nor translation surfaces will be assumed.

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