Affinely prime dynamical systems

Abstract

We study representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently "minimal" actions. When the group in question is PSL(2,R), we exhibit a one-one correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. Our analysis shows that, surprisingly, all these representations are equivalent. In fact we find that all irreducible affine representations of this group are equivalent. The key to this is a property we call "linear Stone-Weierstrass" for group actions on compact spaces, which, if it holds for the "universal strongly proximal space" of the group (to be defined) then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.