This paper deals with 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), the authors exhibit a one-one correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called “linear Stone-Weierstrass” for group actions on compact spaces. 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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Conway, J. B., Functions of one complex variable. II, Graduate Texts in Mathematics, 159, Springer-Verlag, New York, 1995.
Dunford, N. and Schwartz, J., Linear Operators, Part I, 3rd printing, Interscience, New York, 1966.
Furstenberg, H., A Poisson formula for semi-simple Lie groups, Ann. of Math., 77, 1963, 335–386.
Garling, D. J. H., On symmetric sequence spaces, Proc. London Math. Soc., 16(3), 1966, 85–106.
Garling, D. J. H., On ideals of operators in Hilbert space, Proc. London Math. Soc., 17(3), 1967, 115–138.
Glasner, S., Proximal flows, Lecture Notes in Math., 517, Springer-Verlag, New York, 1976.
Lehner, J., Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, RI,1964.
Phelps, R. R., Choquet’s theorem, 2nd edition, Lecture Notes in Mathematics, 1757, Springer-Verlag, Berlin, 2001.
Dedicated to Professor Haim Brezis on the occasion of his 70th birthday
About this article
Cite this article
Furstenberg, H., Glasner, E. & Weiss, B. Affinely prime dynamical systems. Chin. Ann. Math. Ser. B 38, 413–424 (2017). https://doi.org/10.1007/s11401-017-1076-7
- Irreducible affine dynamical systems
- Affinely prime
- Strong proximality
- Möbius transformations
- Harmonic functions
2000 MR Subject Classification