Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. (English) Zbl 0603.58016

De la Llave, R.; Marco, J. M.; Moriyon, R.

Ann. Math. (2) 123, 537-611 (1986).

The goal of this paper is to construct canonical perturbation theory for all Anosov symplectic diffeomorphisms and for all Hamiltonian flows Anosov on each energy surface. In the first section, the authors introduce a formalism to deal with conjugacy equations between families of canonical maps, and then find another geometrically natural way of reducing conjugacy equations to equations between functions. As a consequence of this naturalness, the resulting equations are very simple. In the second section, the authors devote themselves to the study of the regularity of solutions of cohomology equations for Anosov diffeomorphisms and flows, including dependence with respect to parameters, and also with the application of these results to studying the regularity of invariant measures.
There are in addition three Appendices in this paper. In Appendix A, the authors give a proof of the Anosov structural stability theorem for flows, showing that the change of variables and the time change depend in a \(C^{\infty}\) fashion on the perturbation.

Reviewer: Ding Tongren

MSC:

37D99	Dynamical systems with hyperbolic behavior
37J40	Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34E10	Perturbations, asymptotics of solutions to ordinary differential equations

Keywords:

Anosov systems; canonical perturbation; symplectic diffeomorphisms; cohomology equations; regularity of invariant measures