Injectivity as a transversality phenomenon in geometries of negative curvature

The global asymptotic stability conjecture in dynamical systems was solved recently and independently by Feller, Glutsiuk and Gutierrez. Crucial to th e approach of Gutierrez is the following theorem of his: A local diffeomorp hism f: R-2 --> R-2 for which the eigenvalues of Df (x) miss (0, infinity) must be injective. The present paper gives a partial generalization of this theorem to local diffeomorphisms between Hadamard surfaces, the spectral c ondition being replaced by transversality conditions among certain foliatio ns associated to horocycles. The proofs use arguments from global analysis.