Minimality of Strong Foliations of Anosov and Partially Hyperbolic Diffeomorphisms

TL;DR

This work investigates the topological dynamics of expanding invariant foliations for $C^{1+}$ diffeomorphisms in the partially hyperbolic setting with a $1$-dimensional center. It introduces s-transversality, a scale-invariant crossing property for uu-laminations, together with the SH (some hyperbolicity) condition, and proves that an $s$-transverse, SH lamination necessarily contains a $cu$-disk (tangent to $E^{uu} oplus E^c$). This structural result yields a robust criterion for minimality of the strong unstable foliation $\, ext{W}^{uu}$, establishing $C^1$-robust minimality under the non-joint integrability of $E^{uu} oplus E^s$ in the Anosov/partially hyperbolic setting, with applications to ${\\mathbb T}^3$ and toward density questions in future work. The paper develops brushes, holonomies, and plaque frameworks to control center-stable/unstable interactions, and it lays out a detailed program to deduce both openness and density results for minimality via perturbative and topological techniques. The results connect geometric transverse structure with measure-theoretic consequences, such as the uniqueness of certain uu-Gibbs states under non-integrability, highlighting a broad impact on the understanding of dynamics in low-dimensional partially hyperbolic systems.

Abstract

We study the topological properties of expanding invariant foliations of $C^{1+}$ diffeomorphisms, in the context of partially hyperbolic diffeomorphisms and laminations with $1$-dimensional center bundle. In this first version of the paper, we introduce a property we call *s-transversality* of a partially hyperbolic lamination with $1$-dimensional center bundle, which is robust under $C^1$ perturbations. We prove that under a weak expanding condition on the center bundle (called *some hyperbolicity*, or "SH"), any s-transverse partially hyperbolic lamination contains a disk tangent to the center-unstable direction (Theorem C). We obtain several corollaries, among them: if $f$ is a $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center, and the (strong) unstable foliation $W^{uu}$ of $f$ is minimal, then $W^{uu}$ is robustly minimal under $C^1$-small perturbations, provided that the stable and strong unstable bundles are not jointly integrable (Theorem B). Theorem B has applications in our upcoming work with Eskin, Potrie and Zhang, in which we prove that on ${\mathbb T}^3$, any $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center has a minimal strong unstable foliation, and has a unique $uu$-Gibbs measure provided that the stable and strong unstable bundles are not jointly integrable. In a future work, we address the density (in any $C^r$ topology) of minimality of strong unstable foliations for $C^{1+}$ partially hyperbolic diffeomorphisms with $1$-dimensional center and the SH property.

Figures (18)

• Figure 1: Informal definition of s-transversality.
• Figure 2:
• Figure 3: A brush.
• Figure 4: The set $U_\varepsilon(x_0)$.
• Figure 5: Hexagonal loop attached to $\mathcal{W}^{uu}(x)$.

Theorems & Definitions (54)

• Theorem A
• Theorem B
• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Definition 1.6
• Theorem C
• Corollary 1.7