Pathological foliations and removable zero exponents. (English) Zbl 0976.37013

Summary: The ergodic theory of uniformly hyperbolic, or “Axiom A”, diffeomorphisms has been studied extensively, beginning with the pioneering work of Anosov, Sinai, Ruelle and Bowen. While uniformly hyperbolic systems enjoy strong mixing properties, they are not dense among \(C^1\) diffeomorphisms. Using the concept of Lyapunov exponents, Pesin introduced a weaker form of hyperbolicity, which he termed nonuniform hyperbolicity. Nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones. Our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms. As a curious by-product of our construction, we prove that a pathological feature of central foliations – the complete failure of absolute continuity – can exist in a \(C^1\)-open set of volume-preserving diffeomorphisms.


37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C20 Generic properties, structural stability of dynamical systems
37A05 Dynamical aspects of measure-preserving transformations
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