##
**Perturbations of linear quasi-periodic system.**
*(English)*
Zbl 1200.47046

Marmi, Stefano (ed.) et al., Dynamical systems and small divisors. Lectures given at the C. I. M. E. summer school, Cetraro, Italy, June 13–20, 1998. Berlin: Springer (ISBN 3-540-43726-6/pbk). Lect. Notes Math. 1784, 1–60 (2002).

From the text: Existence of both Floquet and \(l^2\) solutions of linear quasi-periodic skew-products can be formulated in terms of linear operators on \(l^2(\mathbb Z)\), i.e., \(\infty\)-dimensional matrices. In the perturbative regime, these matrices are perturbations of diagonal matrices and the problem is to diagonalize them completely or partially, i.e., to show that they have some point spectrum.

The unperturbed matrices have a dense point spectrum so that their eigenvalues are, up to any order of approximation, of infinite multiplicity, which is a very delicate situation to perturb. For matrices with strong decay of the matrix elements off the diagonal, this difficulty can be overcome if the eigenvectors are sufficiently well clustering. One way to handle this is to control the almost multiplicities of the eigenvalues.

The eigenvalues are given by functions of one or several parameters and, in order to control the almost multiplicities, it is necessary that these functions are not too flat. Such a condition is delicate to verify since derivatives of eigenvalues of a matrix behave very badly under perturbations of the matrix. Derivatives of eigenvalues of matrices are therefore replaced by derivatives of resultants of matrices – an object which behaves better under perturbations.

If the parameter space is one-dimensional and if the quasi-periodic frequencies satisfy some Diophantine condition, then it turns out that this control of the derivatives of eigenvalues, in terms of the resultants, is not only necessary but also sufficient for the control of the almost multiplicities. If the parameter space is higher-dimensional, this control is more difficult to achieve and not yet well understood.

In Section 1, we introduce some notations. In Section 2, we define the normal form matrices which are a sort of generalized block matrices. They depend smoothly on some parameters and are covariant with respect to a group action. The smoothness is only piecewise in a way we explain. In Section 3, we show that we can conjugate the normal form matrices to block diagonal form, but there is a price to pay – the smoothness properties of the conjugated matrix is less good. In Section 4, we show that we can conjugate a perturbation of a normal form matrix to a new normal form matrix with a much smaller perturbation. This should be the starting point for an iteration of KAM type.

In Section 5, we discuss under what conditions this procedure can be iterated and the role of clustering of the blocks. Up to this point, neither the smoothness nor the group action play any role whatsoever. The clustering property is related to almost multiplicities of the eigenvalues. These multiplicities are related to estimates of resultants which can be obtained from a transversality property. In Section 6, we discuss when and how this transversality property of the resultants can be transferred to nearby normal form matrices.

In Section 7, we specialize to a quasi-periodic group action on the one-dimensional torus, i.e., to a linear ergodic action on \(T\), submitted to a Diophantine condition. In this case, the transversality property of the resultants will be sufficient to control the almost multiplicities of the eigenvalues and hence the clustering of the blocks. This will give us a perturbation theorem, Theorem 12, of a quite general type. In Section 8, we discuss applications of Theorem 12 to some of the one-dimensional problems we have discussed above.

In an appendix, we include basic finite-dimensional results from analysis and linear algebra.

There is some difference, besides presentation, from the approach in [L.H.Eliasson, Acta Math.179, 153–196 (1997; Zbl 0908.34072)]. We work with Gevrey functions – the particular Gevrey class has no importance – but instead of using the possibility to smoothly truncate Gevrey function as in [loc.cit.], we use here discontinuous cut-offs. The disadvantage is that we have to work with piecewise smooth functions, which is a little awkward in relation to the covariance property, and that we must control the size of the pieces. The advantage is that the particular group action does not intervene before the final step of the proof.

For the entire collection see [Zbl 0989.00040].

The unperturbed matrices have a dense point spectrum so that their eigenvalues are, up to any order of approximation, of infinite multiplicity, which is a very delicate situation to perturb. For matrices with strong decay of the matrix elements off the diagonal, this difficulty can be overcome if the eigenvectors are sufficiently well clustering. One way to handle this is to control the almost multiplicities of the eigenvalues.

The eigenvalues are given by functions of one or several parameters and, in order to control the almost multiplicities, it is necessary that these functions are not too flat. Such a condition is delicate to verify since derivatives of eigenvalues of a matrix behave very badly under perturbations of the matrix. Derivatives of eigenvalues of matrices are therefore replaced by derivatives of resultants of matrices – an object which behaves better under perturbations.

If the parameter space is one-dimensional and if the quasi-periodic frequencies satisfy some Diophantine condition, then it turns out that this control of the derivatives of eigenvalues, in terms of the resultants, is not only necessary but also sufficient for the control of the almost multiplicities. If the parameter space is higher-dimensional, this control is more difficult to achieve and not yet well understood.

In Section 1, we introduce some notations. In Section 2, we define the normal form matrices which are a sort of generalized block matrices. They depend smoothly on some parameters and are covariant with respect to a group action. The smoothness is only piecewise in a way we explain. In Section 3, we show that we can conjugate the normal form matrices to block diagonal form, but there is a price to pay – the smoothness properties of the conjugated matrix is less good. In Section 4, we show that we can conjugate a perturbation of a normal form matrix to a new normal form matrix with a much smaller perturbation. This should be the starting point for an iteration of KAM type.

In Section 5, we discuss under what conditions this procedure can be iterated and the role of clustering of the blocks. Up to this point, neither the smoothness nor the group action play any role whatsoever. The clustering property is related to almost multiplicities of the eigenvalues. These multiplicities are related to estimates of resultants which can be obtained from a transversality property. In Section 6, we discuss when and how this transversality property of the resultants can be transferred to nearby normal form matrices.

In Section 7, we specialize to a quasi-periodic group action on the one-dimensional torus, i.e., to a linear ergodic action on \(T\), submitted to a Diophantine condition. In this case, the transversality property of the resultants will be sufficient to control the almost multiplicities of the eigenvalues and hence the clustering of the blocks. This will give us a perturbation theorem, Theorem 12, of a quite general type. In Section 8, we discuss applications of Theorem 12 to some of the one-dimensional problems we have discussed above.

In an appendix, we include basic finite-dimensional results from analysis and linear algebra.

There is some difference, besides presentation, from the approach in [L.H.Eliasson, Acta Math.179, 153–196 (1997; Zbl 0908.34072)]. We work with Gevrey functions – the particular Gevrey class has no importance – but instead of using the possibility to smoothly truncate Gevrey function as in [loc.cit.], we use here discontinuous cut-offs. The disadvantage is that we have to work with piecewise smooth functions, which is a little awkward in relation to the covariance property, and that we must control the size of the pieces. The advantage is that the particular group action does not intervene before the final step of the proof.

For the entire collection see [Zbl 0989.00040].

### MSC:

47B39 | Linear difference operators |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

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

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

39A12 | Discrete version of topics in analysis |

82B05 | Classical equilibrium statistical mechanics (general) |