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 dieomorphisms. 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 fibers. We prove, in particular, that if the genus of the fiber is at least 2 then the Lyapunov
exponents must be different 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 diffeomorphisms on the fibers. 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.