##
**Reducibility of skew-product systems with values in compact groups.
(Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts.)**
*(French)*
Zbl 0957.37016

Astérisque. 259. Paris: Société Mathématique de France, vi, 216 p. (1999).

The book is devoted to the study of the problem of reducibility (conjugacy to constants) of quasi-periodic discrete skew-product systems with values in compact semisimple groups and to the associated problem of the existence of Floquet type solutions for linear differential quasi-periodic systems with values in compact semisimple algebras.

The main result is that for real one-parameter families of quasi-periodic systems with values in the group of rotations of \(\mathbb R^3\), reducibility holds for almost all values of the parameter in the case that the family is close to some family of constant systems. The proof is based on a resonance removing procedure due to L. H. Eliasson and a notion of Pyartly transversality, which gives a possibility to control the dependence of eigenvalues on the parameter.

Along with the main result many other interesting results on systems of this type are obtained. A brief description of the book’s contents is the following. Chapter 1 is an introduction to the problem of reducibility of skew-product diffeomorphisms and quasi-periodic linear differential equations. Chapter 2 gives necessary and sufficient conditions for reducibility of skew-product systems and presents the study of the centralizer of constant systems. In Chapter 3, a positive measure reducibility theorem is proven for the compact semisimple group case. Chapter 4 is devoted to normal form type theorems and their applications. In Chapter 5 it is shown that in the compact semisimple group case (modulo some finite covering which depends only on the group), the set of reducible systems is dense near the constants. Chapter 6 contains the main result mentioned above.

Clearly and beautifully written the book is highly recommended for those engaged in the study of skew-product and quasi-periodic systems.

The main result is that for real one-parameter families of quasi-periodic systems with values in the group of rotations of \(\mathbb R^3\), reducibility holds for almost all values of the parameter in the case that the family is close to some family of constant systems. The proof is based on a resonance removing procedure due to L. H. Eliasson and a notion of Pyartly transversality, which gives a possibility to control the dependence of eigenvalues on the parameter.

Along with the main result many other interesting results on systems of this type are obtained. A brief description of the book’s contents is the following. Chapter 1 is an introduction to the problem of reducibility of skew-product diffeomorphisms and quasi-periodic linear differential equations. Chapter 2 gives necessary and sufficient conditions for reducibility of skew-product systems and presents the study of the centralizer of constant systems. In Chapter 3, a positive measure reducibility theorem is proven for the compact semisimple group case. Chapter 4 is devoted to normal form type theorems and their applications. In Chapter 5 it is shown that in the compact semisimple group case (modulo some finite covering which depends only on the group), the set of reducible systems is dense near the constants. Chapter 6 contains the main result mentioned above.

Clearly and beautifully written the book is highly recommended for those engaged in the study of skew-product and quasi-periodic systems.

Reviewer: Eugene Ershov (St.Peterburg)

### MSC:

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

34C25 | Periodic solutions to ordinary differential equations |

37K99 | Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems |

37C55 | Periodic and quasi-periodic flows and diffeomorphisms |

37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |