Geodesic flows on closed hyperbolic surfaces are a quintessential example of chaotic dynamics, i.e., systems whose long term behavior is very sensitive to initial conditions. The speed of such chaos is controlled by the spectral gap of the Laplace-Beltrami operator of the underlying hyperbolic surface. In this paper we give an overview of recent breakthroughs of Anantharaman and Monk showing that large genus closed hyperbolic surfaces have optimal spectral gap in a probabilistic sense. On the way we introduce and discuss the foundational works of many authors, from Selberg to Mirzakhani, that play a crucial role in the tour de force proof of Anantharaman and Monk.
We prove the existence of an essential spectral gap for convex cocompact complex hyperbolic manifolds under the hypothesis that the limit set does not contain a complex circle. We apply an extension to complex hyperbolic manifolds due to Quan of the approach of Vasy to meromorphic continuation of the resolvent. This al...
We show that, for Weil--Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessari...
This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from...
We study entropy and periodic-orbit growth for flows on metric spaces. First, we prove that the finite topological entropy of a flow on a compact metric space is completely carried by compact subsets of its regular set. Next, we establish a Bowen--Walters inequality for geometrically separating flows on possibly noncom...
We describe a version of positive macroscopic scalar curvature motivated by the work of Alpert, Balitskiy, and Guth, and prove that this condition on a manifold implies a bound on its 1-width in terms of its first Betti number. A key tool in the proof is a decomposition of any closed manifold into a family of chains wi...
We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3\pi/5, and it is attained precisely by quadratic differentials q=\omega^2 for which the translation surface (X,\omega) lies in the GL_2^+(R)-orbit of t...
Xiaoyu Su, Yu-Min Zhong· 0 citations
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