Fragile cycles

We study diffeomorphisms f with heterodimensional cycles, that is, heteroclinic cycles associated to saddles p and q with different indices. Such a cycle is called fragile if there is no diffeomorphism close to f with a robust cycle associated to hyperbolic sets containing the continuations of p and q. We construct a codimension one submanifold of Diff1(S2×S1) that consists of diffeomorphisms with fragile heterodimensional cycles. Our construction holds for any manifold of dimension ⩾4.