Equivariant hyperbolic diffeomorphisms and representation coverings

TL;DR

This paper establishes a correspondence between G-equivariant hyperbolic diffeomorphisms with convergence properties and G-stable open coverings modeled on tangent G-representations, especially for holomorphic torus actions.

Contribution

It proves a new equivalence linking hyperbolic diffeomorphisms and G-representation coverings for compact G-manifolds with fixed points.

Findings

Existence of G-stable open coverings modeled on tangent representations.

Equivalence between hyperbolic diffeomorphisms and representation coverings for holomorphic torus actions.

Characterization of G-manifolds via equivariant diffeomorphisms and fixed point data.

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