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Density and location of resonances for convex co-compact hyperbolic surfaces

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Abstract

Let \(X=\varGamma\backslash \mathbb {H}^{2}\) be a convex co-compact hyperbolic surface and let δ be the Hausdorff dimension of the limit set. Let Δ X be the hyperbolic Laplacian. We show that the density of resonances of the Laplacian Δ X in rectangles

$$\bigl\{ \sigma\leq \mathrm {Re}(s)\leq\delta,\ \big\vert \mathrm {Im}(s)\big\vert\leq T \bigr\} $$

is less than O(T 1+τ(σ)) in the limit T→∞, where τ(σ)<δ as long as \(\sigma>{\frac {\delta }{2}}\). This improves the previous fractal Weyl upper bound of Zworski (Invent. Math. 136(2):353–409, 1999) and goes in the direction of a conjecture stated in Jakobson and Naud (Geom. Funct. Anal. 22(2):352–368, 2012).

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Notes

  1. We recall that \(\mathcal{R}_{X}\) has at most finitely many points with \(\mathrm {Re}(s) \geq {\frac {1}{2}}\), the largest one being the simple eigenvalue at s=δ.

  2. “Trivial” or topological zeros of Z Γ (s) are located at s=−n, \(n\in \mathbb {N}_{0}\), with mutiplicities 2n+1, and correspond with multiplicities to the resonances of \(\mathbb {H}^{2}\). See Theorem 10.1 in [5] for a precise factorization of Z Γ (s).

  3. The main reason that forces us to perform this “small scale analysis” is the exponential blow-up of phases in the complex domain as |Im(s)|→∞, see Sect. 3, estimate (11).

  4. This identity follows from the orbit equivalence, Lemma 15.3 in [5].

  5. Convexity follows obviously from the variational formula above.

  6. It means that outside a disc of radius \(r_{0}<e^{P(\sigma_{0})}\), the spectrum is made of isolated eigenvalues of finite multiplicity, see [2] Chap. 1, for precise references.

  7. Which is simply obtained by repeated integration by parts in our case.

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Acknowledgements

I would like to thank both referees for valuable remarks that definitely helped to improve the readability of the paper. I wish to thank Colin Guillarmou, Stéphane Nonnenmacher and Maciej Zworski for their reading and comments. I also thank D. Jakobson for our collaboration which motivated this paper.

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  1. Laboratoire d’Analyse non-linéaire et Géométrie (EA 2151), Université d’Avignon et des pays de Vaucluse, 84018, Avignon, France

    Frédéric Naud

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Naud, F. Density and location of resonances for convex co-compact hyperbolic surfaces. Invent. math. 195, 723–750 (2014). https://doi.org/10.1007/s00222-013-0463-2

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