Abstract. Let Γ < SL 2 (Z) be a non-elementary finitely generated subgroup and let Γ(q) be its congruence subgroup of level q for each q ∈ N. We obtain an asymptotic formula for the matrix coefficients of L 2 (Γ(q)\ SL 2 (R)) with a uniform exponential error term for all square-free q with no small prime divisors. As an application we establish a uniform resonance-free half plane for the resolvent of the Laplacian on Γ(q)\H 2 over q as above. Our approach is to extend Dolgopyat's dynamical proof of exponential mixing of the geodesic flow uniformly over congruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingredients is the expander theory due to Bourgain-Gamburd-Sarnak.
Let G be a connected simple linear Lie group of rank one, and let Γ < G be a discrete Zariski dense subgroup admitting a finite Bowen-Margulis-Sullivan measure m BMS . We show that the right translation action of the one dimensional diagonalizable subgroup is mixing on (Γ\G, m BMS ). Together with the work of Roblin, this proves ergodicity of the Burger-Roblin measure under the horospherical group N , establishes a classification theorem for N invariant Radon measures on Γ\G, and provides precise asymptotics for the Haar measure matrix coefficients.
Citation for published item:wgeeD wihel nd yhD ree nd interD hle @PHIUA 9niform ongruene ounting for hottky semigroups in vP@AF9D tournl f¤ ur die reine und ngewndte wthemtikF a grelles journlF F Further information on publisher's website:The nal publication is available at www.degruyter.com With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee.
Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let Γ be a Schottky semigroup in SL2(Z), and for q ∈ N, let Γ(q) := {γ ∈ Γ : γ = e (mod q)} be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls BR in M2(R) of radius R: for all positive integer q with no small prime factors,as R → ∞ for some cΓ > 0, C > 0, > 0 which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of SL2(Z), which arises in the study of Zaremba's conjecture on continued fractions.
Abstract. We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of S 1 . We will show that these zeta functions are nonvanishing on a half plane ℜ(s) > δ−ǫ, where δ is the Hausdorff dimension of the Julia set of f .
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