Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 413–424 | Cite as

Affinely prime dynamical systems

Article
  • 34 Downloads

Abstract

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.

Keywords

Irreducible affine dynamical systems Affinely prime Strong proximality Möbius transformations Harmonic functions 

2000 MR Subject Classification

31A05 37B05 54H11 54H20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Conway, J. B., Functions of one complex variable. II, Graduate Texts in Mathematics, 159, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
  2. [2]
    Dunford, N. and Schwartz, J., Linear Operators, Part I, 3rd printing, Interscience, New York, 1966.MATHGoogle Scholar
  3. [3]
    Furstenberg, H., A Poisson formula for semi-simple Lie groups, Ann. of Math., 77, 1963, 335–386.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Garling, D. J. H., On symmetric sequence spaces, Proc. London Math. Soc., 16(3), 1966, 85–106.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Garling, D. J. H., On ideals of operators in Hilbert space, Proc. London Math. Soc., 17(3), 1967, 115–138.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Glasner, S., Proximal flows, Lecture Notes in Math., 517, Springer-Verlag, New York, 1976.CrossRefGoogle Scholar
  7. [7]
    Lehner, J., Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, RI,1964.CrossRefMATHGoogle Scholar
  8. [8]
    Phelps, R. R., Choquet’s theorem, 2nd edition, Lecture Notes in Mathematics, 1757, Springer-Verlag, Berlin, 2001.MATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Hillel Furstenberg
    • 1
  • Eli Glasner
    • 2
  • Benjamin Weiss
    • 1
  1. 1.Institute of MathematicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Department of MathematicsTel Aviv UniversityTel AvivIsrael

Personalised recommendations