Yang Jianli's Ph.D. Thesis in MathematicsYang Jianli Abstract: To any diffeomorphism we can canonically associate linear operators on the Banach spaces of continuous or bounded sections of the tangent bundle. In many cases dynamical properties of the diffeomorphism, especially hyperbolicity, can be expressed in terms of the spectrum of the associated operator, which is called the adjoint spectrum of the diffeomorphism. Such characterizations are known for Anosov and quasi-Anosov diffeomorphisms, due to J. Mather and R. Mane respectively. M. Brin proves that an Anosov diffeomorphism satisfying a pinching condition on its adjoint spectrum, "pinched Anosov in the sense of Brin", is topologically conjugate to a hyperbolic infranilmanifold automorphism. The adjoint spectrum may also be attached to any orbit or invariant set and is a generalization of the set of eigenvalues of the tangent to the diffeomorphism at a fixed point. In this paper we study many inclusion relations between adjoint spectra attached to various invariant sets and some of their topological properties. We discover that the adjoint spectral radii of a linear toral automorphism are equal to its spectral radii. We show the existence of Brin's pinched Anosov diffeomorphisms of all codimensions, and in particular the existence of Anosov diffeomorphisms of all codimensions. P. McSwiggen has posed the question of the existence of codimension one linear toral Anosov diffeomorphisms satisfying a certain spectral condition. We answer his question in all codimensions. We also give more general characterizations for quasi-Anosov diffeomorphisms, which extend Mane's results. -------------------------- |