Dimension of the Limit Set and the Density of Resonances for Convex Co-compact Hyperbolic Surfaces

The purpose of this paper is to show how the methods of Sjj ostrand for proving the geometric bounds for the density of resonances 28] apply to the case of convex co-compact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110 o angle, with the fundamental domain of its subgroup of direct isometries and the associated Riemann surface. The dimension of the limit set is = 0:70055063 :: :. The convex co-compact surfaces can be deened as the innnite volume nite geometry quotients, X = ?nH 2 , for which the group ? has only hyperbolic elements. The limit set of ?, (?) @H 2 , is classically deened as the set of accumulation points of orbits of hyperbolic elements. As we will recall below it is naturally related to the dynamically deened trapped set, K, of X, K T X n 0, 23],,33]. The convex co-compact condition can also be formulated as saying that the projection of the convex hull of the limit set from H 2 to ?nH 2 is compact. Resonances of X are equivalently given as either the poles of the meromorphic continuation of the resolvent, R X (s) = ((X ? s(1 ? s)) ?1 , the scattering matrix of X, or of the Eisenstein series { see Sect.3 below and 24],,12]. They constitute the natural replacement of the discrete spectral data for problems on non-compact domains. Roughly speaking, if an eigenvalue gives the frequency of a state then the real of part of a resonances describes its frequency and the imaginary part its rate of decay. Hence the resonances close to the real axis are perhaps most interesting as they live the longest. Theorem. If X = ?nH 2 is a nite geometry non-compact surface such that ? has only hyperbolic elements and if R X denotes the set of resonances of X included according to their multiplicities, then for any , 1 2 M. ZWORSKI 0 1, and a; b > 0 we have ] s 2 R X : 1 2 ? Re s < aj Imsj + b ; jsj …