Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T15:29:10.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher cohomology for Abelian groups of toral automorphisms

Published online by Cambridge University Press:  19 September 2008

Anatole Katok  and
Svetlana Katok
Anatole Katok
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: [email protected])
Svetlana Katok
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a complete description of smooth untwisted cohomology with coefficients in ℝl for ℤk-actions by hyperbolic automorphisms of a torus. For 1 ≤ n ≤ k − 1 the nth cohomology trivializes, i.e. every cocycle is cohomologous to a constant cocycle via a smooth coboundary. For n = k a counterpart of the classical Livshitz Theorem holds: the cohomology class of a smooth k-cocycle is determined by periodic data.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 3 , June 1995 , pp. 569 - 592
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Chevalley, C.. Deux theoremes d'arithmetique. J. Math. Soc. Japan 3 (1951), 36–44.Google Scholar
[2]Guillemin, V. and Kazhdan, D.. On the cohomology of certain dynamical systems. Topology 19 (1980), 291–299.Google Scholar
[3]Guillemin, V. and Kazhdan, D.. Some inverse spectral results for negatively curved n-manifolds. Proc. Symp. Pure Math. Amer. Math. Soc. 36 (1980), 153–180.Google Scholar
[4]Hurder, S. and Katok, A.. Differentiability, rigidity and Godbillon—Vey classes for Anosov flows. Publ. Math. IHES 72 (1990), 5–61.Google Scholar
[5]Journé, J.-L.. On regularity problem occuring in connection with Anosov diffeomorphisms. Comm. Math. Phys. 10 (1986), 345–352.Google Scholar
[6]Katok, A. (in collaboration with E. A. Robinson). Constructions in Abstract and Smooth Ergodic Theory. Unpublished notes.Google Scholar
[7]Katok, A. and Spatzier, R.. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ. Math. IHES 79 (1994), 131–156.Google Scholar
[8]Katok, A. and Spatzier, R.. Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math. Res. Letters 1 (1994), 193–202.CrossRefGoogle Scholar
[9]Katok, A. and Spatzier, R.. Invariant measures for higher rank hyperbolic abelian actions. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[10]Katok, S.. Closed geodesies, periods and arithmetic of modular forms. Invent. Math. 80 (1985), 469–480.Google Scholar
[11]Livshitz, A.. Homology properties of Y-systems. Math. Notes USSR Acad. Sci. 10 (1971), 758–763.Google Scholar
[12]de la Llave, R.. Analytic regularity for solutions of Livsic's equation and applications to smooth conjugacy of hyperbolic systems. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[13]de la Llave, R., Marko, J. and Moriyon, R.. Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. Math. 123 (1986), 537–611.Google Scholar
[14]Veech, W.. Periodic points and invariant pseudomeasures for toral endomorphisms. Ergod. Th. & Dynam. Sys. 6 (1986), 449–473.Google Scholar
[15]Weiss, E.. Algebraic Number Theory. Chelsea Publishing Company: New York, NY, 1963.Google Scholar