Injectivity as a transversality phenomenon in geometries of negative curvature
The global asymptotic stability conjecture in dynamical systems was solved recently by Feller, Glutsiuk and Gutierrez. Crucial to the approach of \textit{C. Gutierrez} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 12, No. 6, 627-671 (1995; Zbl 0837.34057)] is the following theorem of his: A local diffeomorphism \(f: \mathbb{R}^2\to \mathbb{R}^2\) for which the eigenvalues of \(Df(x)\) miss \((0,\infty)\) must be injective. The present paper gives a partial generalization of this theorem to local diffeomorphisms between Hadamard surfaces, the spectral conditions being replaced by transversality conditions among certain foliations associated to horocycles. The proofs use arguments from global analysis.
scientific article; zbMATH DE number 1333063