Ergodicity of group actions and spectral gap , applications to random walks and Markov shifts

Let (X,B, ν) be a probability space and let Γ be a countable group of ν-preserving invertible maps of X into itself. To a probability measure μ on Γ corresponds a random walk on X with Markov operator P given by Pψ(x) = ∑ a ψ(ax)μ(a). A powerful tool is the spectral gap property for the operator P when it holds. We consider various examples of ergodic… CONTINUE READING