Complex Analysis, Dynamics and Geometry Seminar

Consider a geometrically finite isometry group of a pinched negatively curved contractible manifold.
There are two natural averaging procedures on this group: averaging with respect to balls in the manifold and taking a finitely supported random walk.

These correspond to two natural measures on the boundary the Patterson-Sullivan measure (which for symetric manifolds is in the Lebesgue measure class) and the harmonic measure which is the limit of convolution powers of the random walk.

These two measures satisfy conformality properties with respect to two metrics on the lattice: the metric induced by the orbit map d and the so called Green metric d_G associated to the random walk, which is quasi-isometric to the word metric.

In turn, they correspond to two measures on the unit tangent bundle (the measure of maximal entropy and the harmonic invariant measure) and closed geodesics on the quotient manifold satisfy two different equidistribution properties with respect to the two measures.

We will show that the harmonic and Patterson-Sullivan measures are singular unless the two metrics are roughly similar: |d-c_1 d_G| Speaker(s): Ilya Gekhtman (University of Toronto)