Geometry MMA Seminar: Weak mixing for billiards in polygons

Speaker: Jonathan Chaika, Institute for Advanced Study

Title:  Weak mixing for billiards in polygons

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, the unit tangent bundle is foliated by invariant surfaces and we know 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). This talk will present two results both concerning a strengthening of ergodicity called ‘weak mixing’:

1) A strengthening of Kerckhoff, Masur and Smillie’s result: There exists a polygon where billiard flow is weakly mixing with respect to the natural volume on the unit tangent bundle. 

2) A classification of the rational polygons where the billiard flow is weakly mixing with respect to the natural area on the invariant surfaces that foliate the unit tangent bundle. 

Open questions will be presented and no previous knowledge of billiards nor translation surfaces will be assumed. This is joint work with Giovanni Forni and Francisco Arana-Herrera.

Time:  Friday, September 19, 2025 – 1:30pm-2:30pm

Place: Exploratory Hall, Room 4208