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| Given a linear differential system with quasiperiodic coefficients, whose fundamental solution is a cocycle over a rotation in a torus, a long studied problem is the reducibility of the system, that is the possibility of conjugating it to a constant one, by a quasiperiodic transformation which preserves the Lyapunov exponents and the rotational properties. In the case of a rationally independent frequency vector, the system can be reducible under arithmetical conditions which ensure that the conjugation is smooth, analytic or Gevrey.
Now if the frequency vector has integer relations (which happens if the system depends on a parameter varying in a torus), one cannot expect the system to be reducible to a constant. Instead, one will look for reducibility to a normal form, that is, to a new system which is constant on the orbits of the base dynamics (a "resonant cocycle"). We give algebraic and arithmetical conditions to ensure analytic conjugacy to such a normal form. |
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