Disintegrations of non-hyperbolic ergodic measures along the center foliation of DA maps

TL;DR

This paper demonstrates that for certain partially hyperbolic diffeomorphisms on the 3-torus, non-hyperbolic ergodic measures have a highly constrained structure along the center foliation, intersecting each leaf in at most two points.

Contribution

It establishes a new property of non-hyperbolic ergodic measures for DA maps, showing their disintegration along the center foliation is limited to at most two points per leaf.

Findings

Non-hyperbolic ergodic measures admit a full measure subset intersecting each center leaf in at most two points.

The result applies to diffeomorphisms homotopic to Anosov on $\

The measure disintegration along the center foliation is highly restricted.

Abstract

We show that each non-hyperbolic ergodic measure of a partially hyperbolic diffeomorphism on $\mathbb{T}^3$ which is homotopic to Anosov admits a full measure subset which intersects each center leaf in at most two points.