Wieczorek–Xie–Jones conjecture for normally hyperbolic compact invariant sets

The compactified system is obtained from a nonautonomous system with autonomous future and past limit systems, and the equilibria of those limit systems correspond to equilibria of the compactified system at the boundary components. For a normally hyperbolic compact invariant set of either limit system, one considers its corresponding compact invariant set in the compactified system and the associated local center-stable, center, or center-unstable manifolds. Wieczorek–Xie–Jones conjecture. The statements for equilibria in Theorems and also hold for general normally hyperbolic compact invariant sets of the future and past limit systems. This extends the stated equilibrium results to periodic orbits, tori, strange attractors, and other normally hyperbolic compact invariant sets; whether the corresponding local invariant-manifold conclusions hold in this generality remains open.