2006/05/22 by Alexander Gorodnik, Gorodnik, Alexander, Amos Nevo +1 · 2 citations
Mathematics · #22D40 #22E30 #28D10 #43A10 #43A90 #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP) #math.DS #math.SP #msc:22D40 #msc:22E30 #msc:28D10 #msc:43A10 #msc:43A90
paper · pdf · doi:10.48550/arxiv.math/0605596
111 pages, Extended appendix on volumes added
arxiv created 2007/12/04 · arxiv updated 2009/12/01
We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports. We demonstrate the new explicit error terms that we obtain by a variety of examples.