Generic H\"older foliations with smooth leaves

TL;DR

This paper studies a space of foliations with smooth leaves and H"older holonomies, showing that generically these foliations are non-absolutely continuous with Dirac measures on leaves, relevant to hyperbolic systems.

Contribution

It introduces a space of foliations with specific regularity and demonstrates generic non-absolute continuity and Dirac measures, advancing understanding of foliations in hyperbolic dynamics.

Findings

Generic foliations are non-absolutely continuous.

Conditional measures are Dirac measures on leaves.

Results are motivated by hyperbolic and partially hyperbolic systems.

Abstract

In this work, we consider a specific space of foliations with $C^1$ leaves and H\"older holonomies of the square $M=[0,1]^2$, with some topology and we show that a generic such foliation is non-absolutely continuous, furthermore, the conditional measures defined by Rokhlin disintegration are Dirac measures on the leaves. This space of foliations is motivated by the foliations that appear in hyperbolic systems and partially hyperbolic systems.