Chinese Annals of Mathematics, Series B

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

Affinely prime dynamical systems



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.


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

2000 MR Subject Classification

31A05 37B05 54H11 54H20 


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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

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