Saddle connections on translation surfaces
- Benjamin Dozier.
- [Stanford, California] : [Stanford University], 2018.
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- 1 online resource.
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- In this thesis I prove several theorems on the distribution and number of saddle connections (and cylinders) on translation surfaces. The first main theorem says that saddle connections become equidistributed on the surface. To state this formally we 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. The theorem is that as R approaches infinity, 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 any rational-angled billiard table, the billiard trajectories that start and end at a corner of the table become equidistributed on the table. The main ingredients in the proof are new results on counting saddle connections whose angle lies in a given interval, and a theorem of Kerckhoff-Masur-Smillie. The second main theorem concerns Siegel-Veech constants, which govern counts of saddle connections averaged over different translation surfaces. We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Combined with results of McMullen and Eskin-Mirzakhani-Mohammadi, this yields the (previously conjectured) convergence of sequences of Siegel-Veech constants associated to Teichmuller curves in genus two. The key technical tool used in the proofs of both the main theorems is a recurrence result for arcs of circles in the moduli space of translation surfaces. This is proved using the "system of integral inequalities'' approach first used by Eskin-Masur for translation surfaces.
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- Submitted to the Department of Mathematics.
- Thesis Ph.D. Stanford University 2018.
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