Absolutely singular dynamical foliations

TL;DR

This paper demonstrates that for a specific open set of partially hyperbolic diffeomorphisms on the 3-torus, Lebesgue measure decomposes into atomic measures along the central foliation, revealing a unique measure-theoretic structure.

Contribution

It establishes the atomic decomposition of Lebesgue measure along central leaves for a class of partially hyperbolic diffeomorphisms, extending previous constructions.

Findings

Lebesgue measure decomposes as atomic measure along central leaves

Validates the phenomenon for an open set of diffeomorphisms

Builds on prior work by Shub and Wilkinson

Abstract

We show that for the C^1-open set of partially hyperbolic diffeomorphisms constructed in (M. Shub and A. Wilkinson, "Pathological foliations and removable zero exponents," Invent. math. 139 (2000) 3, 495-508), Lebesgue measure on the 3-torus decomposes as atomic measure along the leaves of the central foliation.