Equivariant hyperbolic diffeomorphisms and representation coverings

Abstract: Let G be a compact Lie group and X be a compact smooth G-manifold with finitely many G-fixed points. We show that if X admits a G-equivariant hyperbolic diffeomorphism having a certain convergence property, there exists an open covering of X indexed by the G-fixed points so that each open set is G-stable and G-equivariantly diffeomorphic to the tangential G-representation at the corresponding G-fixed point. We also show that the converse is also true in case of holomorphic torus actions