Advertisement

Mathematische Annalen

, Volume 334, Issue 3, pp 693–711 | Cite as

The Ruelle-Sullivan map for actions of ℝ n

  • Johannes KellendonkEmail author
  • Ian F. Putnam
Article

Abstract

The Ruelle Sullivan map for an ℝ n -action on a compact metric space with invariant probability measure is a graded homomorphism between the integer Cech cohomology of the space and the exterior algebra of the dual of ℝ n . We investigate flows on tori to illuminate that it detects geometrical structure of the system. For actions arising from Delone sets of finite local complexity, the existence of canonical transversals and a formulation in terms of pattern equivariant functions lead to the result that the Ruelle Sullivan map is even a ring homomorphism provided the measure is ergodic.

Keywords

Probability Measure Geometrical Structure Equivariant Function Ring Homomorphism Invariant Probability Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, Addison Wesley, 1985Google Scholar
  2. 2.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Springer-Verlag, 1982Google Scholar
  3. 3.
    Connes, A.: C *-algèbres et géométrie différentielle, C.R.A.S. Paris, t. 290, Série A, 599–604 (1980)Google Scholar
  4. 4.
    Connes, A.: Non-Commutative Geometry, Acad. Press, San Diego, 1994Google Scholar
  5. 5.
    Forrest, A.H., Hunton, J.: The cohomology and K-theory of commuting homeomorphisms of the Cantor set. Ergod. Th. and Dynam. Sys. 19, 611–625 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Forrest, A.H., Hunton, J., Kellendonk, J.: Topological invariants for projection method patterns, Memoirs of the AMS 159(758), x+120 pp (2002)Google Scholar
  7. 7.
    Glasner, S.: Proximal flows, Lecture notes in math, vol 517 Springer, 1976Google Scholar
  8. 8.
    Kellendonk. J.: Pattern-equivariant functions and cohomology. J. Phys. A 36, 1–8 (2003)CrossRefGoogle Scholar
  9. 9.
    Lindenstrauss,E.: Pointwise theorems for amenable groups, Invent. Math. 146, 259–295 (2001)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Moody R.V.: Meyer Sets and their Duals, in The Mathematics of Long-Range Aperiodic Order, pages 403–441, editor R.V.Moody Kluwer Academic Publishers, 1997Google Scholar
  11. 11.
    Moore C.C., Schochet,C.: Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, 9, Springer-Verlag, New York, 1988Google Scholar
  12. 12.
    Solomyak,B.: Spectrum of Dynamical Systems Arising from Delone Sets, in Quasicrystals and Discrete Geometry, pages 265–275, editor J. Patera, Fields Institute Monographs, AMS, 1998Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Institute Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne
  2. 2.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

Personalised recommendations