The Liouville property and Ergodicity on the boundary

A riemannian manifold has the Liouville property if every bounded harmonic function is constant. We discuss some old and new results about locally symmetric spaces (for example hyperbolic manifolds) that have this property.  In particular, we will discuss some recent counterexamples to a conjecture attributed to Margulis, constructing (higher rank and simple) locally symmetric spaces of infinite co-volume with the Liouville property as well as examples of two quasi-isometric locally symmetric spaces such that one has non-trivial bounded harmonic functions, while the other has none. 

(Based on work in progress with Subhadip Dey and Mikolaj Fraczyk)

Bio
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