Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits

preprint2026arXivOpen access

For every flat surface, almost every flat surface in its $\mathsf{SL}(2,\mathbb{R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.

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Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits

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Uniform distribution of saddle connection lengths in all SL(2,R)\mathsf{SL}(2,\mathbb{R})SL(2,R) orbits

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Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits