On the reducibility of a class of finitely differentiable quasi-periodic linear systems

In this paper, we consider the following system x ˙ = ( A + ε Q ˜ ( t ) ) x , where A is a constant matrix with different eigenvalues, and Q ˜ ( t ) is quasi-periodic with frequencies ω 1 , ω 2 , … , ω r . Moreover, Q ( θ ) = Q ( ω t ) = Q ˜ ( t ) has continuous partial derivatives ∂ b Q ∂ θ j b for j = 1 , 2 , … , r , where b > 9 4 r + 1 ∈ Z , and the moduli of continuity of ∂ b Q ∂ θ j b satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. Under suitable hypothesis of non-resonance conditions and non-degeneracy conditions, we prove that for most sufficiently small ε, the system can be reducible to a constant coefficient differentiable equation by means of a quasi-periodic homeomorphism.