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Higher cohomology for Abelian groups of toral automorphisms

  • Anatole Katok (a1) and Svetlana Katok (a2)

We give a complete description of smooth untwisted cohomology with coefficients in ℝl for ℤk-actions by hyperbolic automorphisms of a torus. For 1 ≤ nk − 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.

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[1]Chevalley C.. Deux theoremes d'arithmetique. J. Math. Soc. Japan 3 (1951), 3644.
[2]Guillemin V. and Kazhdan D.. On the cohomology of certain dynamical systems. Topology 19 (1980), 291299.
[3]Guillemin V. and Kazhdan D.. Some inverse spectral results for negatively curved n-manifolds. Proc. Symp. Pure Math. Amer. Math. Soc. 36 (1980), 153180.
[4]Hurder S. and Katok A.. Differentiability, rigidity and Godbillon—Vey classes for Anosov flows. Publ. Math. IHES 72 (1990), 561.
[5]Journé J.-L.. On regularity problem occuring in connection with Anosov diffeomorphisms. Comm. Math. Phys. 10 (1986), 345352.
[6]Katok A. (in collaboration with E. A. Robinson). Constructions in Abstract and Smooth Ergodic Theory. Unpublished notes.
[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), 131156.
[8]Katok A. and Spatzier R.. Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math. Res. Letters 1 (1994), 193202.
[9]Katok A. and Spatzier R.. Invariant measures for higher rank hyperbolic abelian actions. Ergod. Th. & Dynam. Sys. To appear.
[10]Katok S.. Closed geodesies, periods and arithmetic of modular forms. Invent. Math. 80 (1985), 469480.
[11]Livshitz A.. Homology properties of Y-systems. Math. Notes USSR Acad. Sci. 10 (1971), 758763.
[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.
[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), 537611.
[14]Veech W.. Periodic points and invariant pseudomeasures for toral endomorphisms. Ergod. Th. & Dynam. Sys. 6 (1986), 449473.
[15]Weiss E.. Algebraic Number Theory. Chelsea Publishing Company: New York, NY, 1963.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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