Cohomological equations of Riemannian flows and the Anosov diffeomorphism. (Équations cohomologiques de flots riemanniens et de difféomorphismes d’Anosov.) (French) Zbl 1134.37001

Dehghan-Nezhad, Akbar; El Kacimi Alaoui, Aziz

J. Math. Soc. Japan 59, No. 4, 1105-1134 (2007).

On an oriented connected \(C^\infty\) manifold M endowed with a diffeomorphism \(\gamma\), let \(g\in C^\infty (M)\) and \(X \in \Gamma^{\infty} (M)\). If there exists \(f\in C^{\infty} (M)\) such that \(f-f\circ\gamma = g\) (resp. \(Xf =g\)), then \(f\) is called a solution of the above discrete (resp. continuous) cohomological equation DCE (resp. CCE). The leafwise cohomology of a complete Riemannian Diophantine flow is computed here. Solving the DCE of \((M,\gamma)\) is equivalent to solve the CCE of the manifold endowed with the vector field obtained by the suspension of \((M,\gamma)\). The authors solve explicitly the DCE for the Anosov diffeomorphism on the torus \(\mathbb T^n\) defined by a hyperbolic and diagonalizable matrix \(A\in \text{SL}(n,\mathbb Z)\), whose eigenvalues are all some real positive numbers. This is then use to solve the CCE of the Anosov flow \(\mathcal F\) on the hyperbolic torus \({\mathbb T_A}^{n+1}\) obtained from \(A\) by suspension. Therefore some other geometrical objects associated to \(A\) and \(\mathcal F\), like invariant distributions and the leafwise cohomology are computed here.

Reviewer: Cornelia-Livia Bejan (Iaşi)

MSC:

37A05	Dynamical aspects of measure-preserving transformations
37C10	Dynamics induced by flows and semiflows
58A30	Vector distributions (subbundles of the tangent bundles)
53C12	Foliations (differential geometric aspects)
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Keywords:

invariant distributions; Anosov flows