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Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves
- Publication Year :
- 2022
-
Abstract
- In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface $X$ of genus $g$ is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of $X$. In this survey we give a detailed account of Mirzakhani's proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.<br />Comment: 34 pages, 17 figures
- Subjects :
- Mathematics - Dynamical Systems
Mathematics - Geometric Topology
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2202.04156
- Document Type :
- Working Paper