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Affinely prime dynamical systems

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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.

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Author information

Correspondence to Hillel Furstenberg.

Additional information

Dedicated to Professor Haim Brezis on the occasion of his 70th birthday

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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

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Keywords

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

2000 MR Subject Classification

  • 31A05
  • 37B05
  • 54H11
  • 54H20