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Equidistribution of saddle connections on translation surfaces

Supported in part by NSF grant DGE-114747.
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  • 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.

    Mathematics Subject Classification: Primary: 37E35; Secondary: 32G15.

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  • Figure 1.  Saddle connections of length at most $R = 7$ on a genus two translation surface (opposite sides are identified), in units where the height of the figure is approximately 2. The thickness of each saddle connection is drawn inversely proportional to its length (so the total amount of "paint" used to draw a saddle connection is independent of its length). This choice of thickness is meant to represent the measures $\mu_s$, which are all probability measures, in Theorem 1.1. That theorem says that, as the length bound $R$ goes to infinity, the picture will be uniformly colored. This picture was generated with the help of Ronen Mukamel's $\texttt{triangulated\_surfaces}$ SAGE package.

    Figure 2.  Opposite sides of the polygon are identified to give a genus two translation surface. A cylinder is shown, together with a long saddle connection contained in that cylinder.

    Figure 3.  Regions used in proof of Lemma 2.2

    Figure 4.  Proof of Lemma 4.1. The red points are group $A_1$, while the blue are $A_2$.

    Figure 5.  Adding a saddle connection to a complex, in proof of Proposition 5.4.

    Figure 6.  Comparing averages for Lemma 5.8 (Shadowing)

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