RATIONAL MODE LOCKING FOR HOMEOMORPHISMS OF THE 2-TORUS

In this paper we consider homeomorphisms of the torus R2/Z2, homotopic to the identity, and their rotation sets. Let f be such a homeomorphism, f ~ : R 2 → R 2 be a fixed lift and ρ ( f ~ ) ⊂ R 2 be its rotation set, which we assume to have interior. We also assume that the frontier of ρ ( f ~ ) contains a rational vector ρ ∈ Q2 and we want to understand how stable this situation is. To be more precise, we want to know if it is possible to find two different homeomorphisms f1 and f2, arbitrarily small C0-perturbations of f, in a way that ρ does not belong to the rotation set of f ~ 1 but belongs to the interior of the rotation set of f ~ 2 , where f ~ 1 and f ~ 2 are the lifts of f1 and f2 that are close to f ~ . We give two examples where this happens, supposing ρ = (0, 0). The first one is a smooth diffeomorphism with a unique fixed point lifted to a fixed point of f ~ . The second one is an area preserving version of the first one, but in this conservative setting we only obtain a C0 example. We also present two theorems in the opposite direction. The first one says that if f is area preserving and analytic, we cannot find f1 and f2 as above. The second result, more technical, implies that the same statement holds if f belongs to a generic one parameter family (ft)t∈[0,1] of C2-diffeomorphisms of T2 (in the sense of Brunovsky). In particular, lifting our family to a family ( f ~ t ) t ∈ [ 0 , 1 ] of plane diffeomorphisms, one deduces that if there exists a rational vector ρ and a parameter t* ∈ (0, 1) such that ρ ( f ~ t * ) has non-empty interior, and ρ ∉ ρ ( f ~ t ) for t < t* close to t*, then ρ ∉ int ( ρ ( f ~ t ) ) for all t > t* close to t*. This kind of result reveals some sort of local stability of the rotation set near rational vectors of its boundary.