The Bianchi IX attractor in modified gravity

We consider vacuum anisotropic spatially homogeneous models in certain modified gravity theories [such as Hořava–Lifshitz, λ–R or f(R) gravity], which are expected to describe generic spacelike singularities for these theories. These models perturb the well-known Bianchi models in general relativity (GR) by a parameter v ∈ (0, 1) with GR recovered at v = 1/2. We prove an analogue of the well-known Ringström attractor theorem in GR to the supercritical theories: for any v ∈ (1/2, 1), all solutions of Bianchi type IX converge to an analogue of the Mixmaster attractor, consisting of Bianchi type I solutions (Kasner states) and heteroclinic chains of Bianchi type II solutions. In contrast to GR, there are no solutions that converge to a different set other than the Mixmaster (such as the locally rotationally symmetric solutions in GR).