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 $G$-invariant 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.