Continuidade absoluta de folheações e implicações dinâmicas

Abstract

The main object of study in this work is the absolute continuity property of invariant foliations for dynamical systems. We begin by presenting the dynamical construction of a classical example of a foliation that does not possess this property [13]. Among the various dynamical implications of the absolute continuity of the stable and unstable foliations of partially hyperbolic diffeomorphisms of class C2, we highlight the ergodicity of conservative Anosov diffeomorphisms, also of class C2 [2]. Finally, in the context of partially hyperbolic diffeomorphisms, we investigate the relationship between the central Lyapunov exponents and the absolute continuity of the central foliation, exploring a generalization of the well-known Mañé argument, as presented in the work of Pesin and Hirayama. This latter work leads to the conclusion that, in certain settings within the partially hyperbolic framework, the absolute continuity property fails generically [7].