Reducibility of linear equations with quaziper coeff--survey

1.1 Linear equations with time-depending coefficients......Page 5
1.2 Quasi-periodic motions and quasi-periodic functions......Page 7
1.3 Linear equations with quasi-periodic coefficients and reducibility......Page 10
1.4 Example 1: One frequency and many dimensions. Floquet theory for periodic systems......Page 11
1.5 Example 2: Many frequencies and one dimension. \ Small divisor problems......Page 13
2.1 Linear skew-product flows and vector bundles......Page 17
2.2 The Sacker-Sell spectrum......Page 23
2.3 Relation with Lyapunov exponents......Page 26
2.4 Smoothness of spectral subbundles......Page 27
2.5 A Reducibility Theorem......Page 28
3.1 Introduction. Useful transformations......Page 33
3.2.1 Statement of the problem......Page 35
3.2.2 Some spectral theory of self-adjoint operators......Page 36
3.2.3 Some Spectral theory of 1D Schrödinger operators......Page 42
3.2.4 Some spectral theory of 1D Schrödinger operators with quasi-periodic potential......Page 51
3.3 Ergodic invariants. The rotation number and the upper Lyapunov exponent......Page 58
3.3.1 The rotation number for real potential and real......Page 59
3.3.2 Extension to complex  and relation with Weyl's m-functions......Page 66
3.3.3 The spectral functions for the half and whole line......Page 69
3.3.4 The real part of w. The Lyapunov exponents for Schrödinger equation with quasi-periodic potential......Page 72
3.3.5 Application to Cantor spectrum......Page 77
3.4.1 Reducibility in the re-sol-vent set. Uni-form-ly hyper-bo-lic re-du-ci-bi-li-ty......Page 78
3.4.2 Reducibility in the spectrum. KAM techniques......Page 80
3.4.3 Some more results and applications......Page 93
3.5 Some remarks on non-re-du-ci-bi-li-ty in Schrö-din-ger e-qua-tion with qua-si-pe-rio-dic po-ten-tial......Page 94
4.1 Reducibility for general linear differential equations with quasi-periodic coefficients......Page 99
4.1.1 Idea of proof......Page 104
4.2 Reducibility in compact groups......Page 107
4.2.1 Set up......Page 108
4.2.2 Sketch of proof in the case of SO(3,R)......Page 113
4.2.3 Remarks on the general compact case......Page 119
4.2.4 Two results on non-reducibility in compact groups......Page 120