Seminar 9.17.15 Brown

Speaker: Aaron Brown (University of Chicago)

Title: From entropy to rigidity: applications to lattice actions

Abstract: Consider a smooth action of a lattice in a higher-rank, simple Lie group G on a compact manifold M.  We show that if the dimension of M is sufficiently small relative to the rank of G, then there always exists an invariant probability measure for the action.  If the dimension of M falls in an intermediate range (relative to the rank of G) we show there exists a quasi-invariant measure such that the action is isomorphic to a relatively measure-preserving extension over a standard boundary action.  The proofs of these results follow from existing measure rigidity techniques combined with a new entropy formula for measures invariant under smooth actions of higher-rank Abelian groups.  This formula establishes a “product structure” (along coarse Lyapunov foliations) of entropy for measures invariant under a smooth action of a higher-rank abelian group. The product structure of entropy follows, in turn, from a generalization of the Ledrappier-Young entropy formula to “entropy subordinated to a foliation.”