Discrete conformal groups and measurable dynamics
Bulletin of the American Mathematical Society
Motivated by his study of 2nd order differential equations a(z)w" + b(z)W + c(z)w = 0 Poincaré (1882) unveiled the vast subject of discrete subgroups of conformai transformations, (z -»(az + b)/(cz + d)} 9 their associated Riemann surfaces, and the intricacies of the limit set-Cantor sets, nowhere differentiable curves, etc. In this paper we discuss an interplay between discrete conformai groups acting on any dimensional ball B d+l and measurable dynamics. The development is closely related to
... closely related to ideas considered by Poincaré, for example, (i) the Poincaré series 2 y€Er |Y'.x 0 | 5 where |y'x| is the linear distortion of the Euclidean metric by the conformai transformation y and x 0 lies in the interior ofB d+l . (ii) The interpretation of interior B d+l with its group of conformai transformations as the Poincaré model of non-Euclidean or hyperbolic geometry, H d+l . The dynamics we consider take place on the df-sphere = dB d+l == the visual sphere at oo for H d+l . There are basically three parts to the discussion. Part I. The ergodic properties of the geodesic flow on the associated hyperbolic manifold H d+1 /T such as (i) the excursion pattern of random geodesies into the cuspidal ends of finite volume noncompact hyperbolic manifolds. (ii) the ergodicity of the geodesic flow relative to conformai measures and the divergence of the Poincaré series at the critical exponent s = 8(T). Part II. The ergodic properties of T acting on the tangent spaces to the Poincaré recurrent part of S d . Part III. An interrelation between the critical exponent 8(T) and other quantities such as (i) the Hausdorff dimension of the limit set Ac^, (ii) the "square root" of the lowest eigenvalue of the Laplacian acting on L 2 (H rf+l /T), (iii) the entropy of the geodesic flow, (iv) general Riemannian manifolds. We will state theorems and give ideas and references for the proofs. The discussion of Part I works for the usual Lebesgue measure on S d as well as for any "conformai measure" JU. on S d , that is a finite positive measure JU.