Asymptotic
게시글 주소: https://orbi.kr/00061167026
Let $X$ be a complete hyperbolic surface of finite area and $c_X(L)$ be the number of primitive closed geodesic of length $\leq L$ on $X$. Then
1. (Delsarte, Huber, Selberg) $c_X(L)\sim e^L/L$ as $L\to\infty$.
Let $\mathcal{M}_{g,n}$ be the moduli space of completely hyperbolic Riemann surface of genus $g$ with $n$ cusps. Let $s_X(L)$ be the number of simple closed geodesics of length $\leq L$. Then
2. (Maryam Mirzakhani) For fixed $X\in\mathcal{M}_{g,n}$, $s_X(L)\sim n(X)L^{6g-6+2n}$ as $L\to\infty$ where $n:\mathcal{M}_{n,g}\to\Bbb R_+$ is a proper continuous function.
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"I think it's rarely about what you actually learn in class . . . it's mostly about things that you stay motivated to go and continue to do on your own." - Mirzakhani, M
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