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I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big picture. All the surveys I can find on the subject try to show things in an explicit way, but I'd rather see an abstration. I feel the way dynamicist tacle this problem post Mirzakhani is by giving a flow on $M_{g,n}$, the Teichmüller geodesic flow. By the work of Mirzakhani, this is a ergodic flow. My question is how does this help us understand the function $$f_L:M_{g,n}\rightarrow \mathbb{R}, \quad X\mapsto s_X(L):=\{ \gamma \mid \ell(\gamma)\leq L\}?$$ I feel that estimates of this function should be related to the Ergodic Theorem, but I can not find anything on that.

PS: I'm sorry for the tags, but I'm uncertain which tags I should use.

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    $\begingroup$ It may be that the 2-page Introduction of this paper could help? Cahn, Patricia, Federica Fanoni, and Bram Petri. "Mapping class group orbits of curves with self-intersections." Israel Journal of Mathematics 223, no. 1 (2018): 53-74. arXiv abstract. $\endgroup$
    – Joseph O'Rourke
    Commented May 22, 2020 at 0:15

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