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Higher cohomology for Abelian groups of toral automorphisms

  • Anatole Katok (a1) and Svetlana Katok (a2)
  • DOI:
  • Published online: 01 September 2008

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]C. Chevalley . Deux theoremes d'arithmetique. J. Math. Soc. Japan3 (1951), 3644.

[2]V. Guillemin and D. Kazhdan . On the cohomology of certain dynamical systems. Topology 19 (1980), 291299.

[3]V. Guillemin and D. Kazhdan . Some inverse spectral results for negatively curved n-manifolds. Proc. Symp. Pure Math. Amer. Math. Soc. 36 (1980), 153180.

[4]S. Hurder and A. Katok . Differentiability, rigidity and Godbillon—Vey classes for Anosov flows. Publ. Math. IHES 72 (1990), 561.

[7]A. Katok and R. Spatzier . First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ. Math. IHES 79 (1994), 131156.

[8]A. Katok and R. Spatzier . Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math. Res. Letters 1 (1994), 193202.

[10]S. Katok . Closed geodesies, periods and arithmetic of modular forms. Invent. Math. 80 (1985), 469480.

[11]A. Livshitz . Homology properties of Y-systems. Math. Notes USSR Acad. Sci. 10 (1971), 758763.

[13]R. de la Llave , J. Marko and R. Moriyon . Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. Math. 123 (1986), 537611.

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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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