Bowen{margulis and Patterson Measures on Negatively Curved Compact Manifolds

We give a new simple construction of the maximal entropy invariant measure (the Bowen{Margulis measure) for the geodesic ow on a compact negatively curved manifold. This construction directly connects invariant`conformal density" arising from the Patterson measure on the sphere at innnity of the universal covering space with the maximal entropy measureof the geodesic ow. 0. Introduction Let M be a simply connected negatively curved manifold, @M { its sphere at innnity. Then every innnite geodesic on M has two endpoints on @M, and, conversely, every pair of distinct points from @M determines an innnite geodesic on M. Thus there exists a natural one-to-one correspondence between the Radon invariant measures of the geodesic ow on the unit tangent bundle SM and the Radon measures on the locally compact space @ 2 M of distinct pairs of points from @M. If N is a negatively curved manifold with the fundamental group G = 1 (N), and M { its universal covering space, then Radon invariant measures of the geodesic ow on SM correspond to those invariant measures of the geodesic ow on SM, which are also G-invariant. Hence we have a correspondence between invariant Radon measures of the geodesic ow on SN (or, simply, nite measures, if N is compact) and G-invariant Radon measures on @ 2 M (geodesic currents in terminology of 2)).