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.
Highlights
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
Our results mainly focus on the comprehension of disintegration of volume measure along the central invariant foliation of partially hyperbolic diffeomorphisms
We prove a general result on the disintegration of Lebesgue measure along the central foliation of derived from Anosov diffeomorphisms on T3
Summary
Let (M, μ, B) be a probability space, where M is a compact metric space, μ a probability measure and B the borelian σ-algebra. The disintegration of a measure along a general foliation is defined in compact foliated boxes, it makes sense to say that the foliation F has a quantity k0 ∈ N atoms per leaf. Definition 2.1 shows that it only make sense to talk about conditional measures from the generic point of view, when restricted to a foliated box B, the set A ∩ B has μ-full measure on B, the support of the conditional measure disintegrated on B must be contained on the set A. For a more detailed discussion about dependence of disintegration to the foliated box see Lemma 3.2 of [1]
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