Abstract
Fix a translation surface $ X $, and consider the measures on $ X $ coming from averaging the uniform measures on all the saddle connections of length at most $ R $. Then, as $ R\to\infty $, the weak limit of these measures exists and is equal to the area measure on $ X $ coming from the flat metric. This implies that, on a rational-angled billiard table, the billiard trajectories that start and end at a corner of the table are equidistributed on the table. We also show that any weak limit of a subsequence of the counting measures on $ S^1 $ given by the angles of all saddle connections of length at most $ R_n $, as $ R_n\to\infty $, is in the Lebesgue measure class. The proof of the equidistribution result uses the angle result, together with the theorem of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.
Highlights
Consider a polygonal billiard table, with a frictionless billiard ball bouncing around inside according to the rule that, when it hits a side, angle of incidence equals angle of reflection
This is a much studied dynamical system that exhibits rich and complicated behavior, and it can be seen as a model for basic physical systems, such as a gas molecule bouncing around in a box
When the ball hits a corner of the table, its future trajectory is not well-defined, and this presence of singular trajectories is in some sense responsible for the complicated behavior of the system
Summary
Consider a polygonal billiard table, with a frictionless billiard ball bouncing around inside according to the rule that, when it hits a side, angle of incidence equals angle of reflection. The main result of this paper (Theorem 1.1) is that the set of saddle connections becomes equidistributed on any translation surface. Let X be a translation surface, and let νR be the probability measure on S1 given by normalized counting measure on the angles of saddle connections of length at most R on X. A somewhat weaker statement than Theorem 1.5, namely a quadratic lower bound for which the constant c1 depends on the surface X, was proven by Marchese-Trevino-Weil ([MTW16], Theorem 1.9, item (4) and Proposition 4.5; they make use of the main results of [Cha11]). We will use the following basic, well-known fact, which gives a universal upper bound on the length of the shortest saddle connection. For any unit-area translation surface X in any stratum H, we have the following universal bound on the length of the shortest saddle connection (X):. Since ν-a.e. fiber measure is area measure, η is area measure
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