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Creators/Authors contains: "ARANA-HERRERA, FRANCISCO"

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  1. (, Ergodic Theory and Dynamical Systems)
    Abstract In her thesis, Mirzakhani showed that the number of simple closed geodesics of length$$\leq L$$on a closed, connected, oriented hyperbolic surfaceXof genusgis asymptotic to$$L^{6g-6}$$times a constant depending on the geometry ofX. 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. 
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    Free, publicly-accessible full text available February 24, 2026
  2. ; (, Geometry & Topology)
    Full Text Available 
  3. (, International Mathematics Research Notices)
    Abstract We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any irreducible $$\textrm {SL}{(2,\textbf {R})}$$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems. 
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  4. ; (, Forum of Mathematics, Sigma)
    null (Ed.)
    Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics. 
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