Journal of Modern Dynamics (JMD)

Minimal yet measurable foliations

Pages: 93 - 107, Issue 1, March 2014      doi:10.3934/jmd.2014.8.93

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Gabriel Ponce - Departamento de Matemática, ICMC-USP São Carlos- SP, Brazil (email)
Ali Tahzibi - Departamento de Matematica, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São Carlos-SP, Brazil (email)
Régis Varão - Departamento de Matemática, ICMC-USP São Carlos-SP, Brazil (email)

Abstract: In this paper we mainly address the problem of disintegration of Lebesgue measure along the central foliation of volume-preserving diffeomorphisms isotopic to hyperbolic automorphisms of 3-torus. We prove that atomic disintegration of the Lebesgue measure (ergodic case) along the central foliation has the peculiarity of being mono-atomic (one atom per leaf). This implies the measurability of the central foliation. As a corollary we provide open and nonempty subset of partially hyperbolic diffeomorphisms with minimal yet measurable central foliation.

Keywords:  Partial hyperbolicity, minimal foliations, measurable partitions, derived from Anosov, disintegration, Lyapunov exponents.
Mathematics Subject Classification:  Primary: 37D30, 37D25; Secondary: 28D99, 37D10.

Received: August 2013;      Available Online: July 2014.