Periodic solutions with prescribed minimal period of vortex type problems in domains

We consider Hamiltonian systems with two degrees of freedom of point vortex type Κ j ?Ζj=J▿ Ζ jHΩ(Ζ 1,Ζ2),j=1,2,for(Ζ1,(Ζ2 in a domain Ω ⊂ ℝ2. In the classical point vortex context the Hamiltonian HΩ is of the form HΩ (Ζ 1, Ζ2)=Κ1)Κ2)log|Ζ1)Ζ2)|2Κ 1) Κ 2g(Ζ 1, Ζ 2x,)Κ 21 h(Ζ 1)Κ 22h(Ζ 2 where g : Ω× Ω → R is the regular part of a hydrodynamic Green function in Ω, h : Ω → R is the Robin function: h(Ζ) = g(Ζ, Ζ), and Κ 1, Κ 2 are the vortex strengths. We prove the existence of infinitely many periodic solutions with prescribed minimal period that are superpositions of a slow motion of the center of vorticity close to a star-shaped level line of h and of a fast rotation of the two vortices around their center of vorticity. The proofs are based on a recent higher dimensional version of the PoincaréBirkhoff theorem due to Fonda and Ureña.