2015/12/16 by Hillel Furstenberg, Hillel Fürstenberg, Eli Glasner +4
Mathematics · #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DS
paper · pdf · doi:10.48550/arxiv.1512.05087
arxiv created 2015/12/16 · openalex publication_date 2015/12/16 · arxiv updated 2015/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.