Preprint A606/2008
Extremal Lyapunov exponents: an invariance principle and applications
Marcelo Viana | Avila, Artur
Keywords: Lyapunov exponent | cocycle | holonomy map
We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic ('center') behavior, e.g. partially hyperbolic di eomorphisms. A number of applications illustrate its power. We find that any ergodic automorphism of the 4-torus with two eigenvalues in the unit circle is stably Bernoulli among symplectic maps. Indeed, any nearby symplectic map has no zero Lyapunov exponents, unless it is volume preserving conjugate to the automorphism itself. Another main application is to accessible skew-product maps preserving area on the fi bers. We prove, in particular, that if the genus of the fiber is at least 2 then the Lyapunov exponents must be di fferent from zero and vary continuously with the map. These, and other dynamical conclusions, originate from a general Invariance Principle we prove in here. It is formulated in terms of smooth cocycles, that is, fiber bundle morphisms acting by diff eomorphisms on the fi bers. The extremal Lyapunov exponents measure the smallest and largest exponential rates of growth of the derivative along the fibers. The Invariance Principle states that if these two numbers coincide then the fibers carry some amount of structure which is transversely invariant, that is, invariant under certain canonical families of homeomorphisms between fibers.