Ergodic Theory and Dynamical Systems

Research Article

Higher cohomology for Abelian groups of toral automorphisms

Anatole Katoka1 and Svetlana Katoka2

a1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: katok_a@math.psu.edu)

a2 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (e-mail: katok_s@math.psu.edu)

Abstract

We give a complete description of smooth untwisted cohomology with coefficients in l for k-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.

(Received December 12 1993)

(Revised June 28 1994)

Footnotes

† The work of the authors was partially supported by NSF grants DMS-9017995 and DMS-9207728, respectively.