On the finite cyclicity of open period annuli

Let Π be an open, relatively compact period annulus of real analytic vector field X0 on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from Π under a given multiparameter analytic deformation Xλ of X0 is finite, provided that X0 is either Hamiltonian, or generic Darbouxian vector field. 1 Statement of the result Let S be a real analytic surface without border (compact or not), and X0 a real analytic vector field on it. An open period annulus of X0 is an union of period orbits of X0 which is bi-analytic to the standard annulus S × (0, 1), the image of each circle S × {u} being a periodic orbit of X0. Let Xλ, λ ∈ (R, 0) be an analytic family of analytic vector fields on S, and let Π be an open period annulus of X0. The cyclicity Cycl(Π, Xλ) of Π with respect to the deformation Xλ is the maximal number of limit cycles of Xλ which tend to Π as λ tends to zero, see Definition 2 bellow. Clearly the vector field X0 has an analytic first integral f in the period annulus Π which