On the indices of periodic points in $C^1$-generic wild homoclinic classes in dimension three

We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that if such a homoclinic class contains a volume-expanding periodic point, then, $C^1$-generically, it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.