The spectrum of the adjoint representation and the hyperbolicity of dynamical systems

Abstract

Let (M,g) denote a smooth compact Riemannian manifold. Anosov [I] defined the global hypcrbolicity of a C2 diffeomorphism (resp. a nonsingular flow f’) on M :f (resp. fr) is h yperbolic or, as we shall say, Anosov, if the tangent bundle TM splits as a sum of invariant subbundles TM = E’ ~3 E(resp. TM = B @ E@ [A’, where [X7 is th I e ine bundle spanned by S = d/dt(ft)) such that Tf exponentially contracts E+ in positive time while exponentially contracting Ein negative time. In this paper we exhibit several criteria which are equivalent to Anosov’s hyperbolicity condition. Our work continues the studies of a number of authors, which we can describe after fixing some terminology. Let C(E) denote the real Banach space of continuous sections of TM or, in the flow case, sections of TM/[X] with norm ),I 17 1) = sup(g,(~, 7) / .Y E M}. Define the adjoint representation f* off (resp. fi off’) as the bounded operator on C(E):