Anosov flows in dimension 3 from gluing building blocks with quasi-transverse boundary

TL;DR

This work develops a comprehensive gluing framework to construct Anosov flows in dimension 3 by assembling flexible building blocks with quasi-transverse boundaries. It introduces normalization tools (affine sections, multipliers, and affine foliations), analyzes the crossing map between entrance and exit boundaries, and designs a spreading mechanism via boundary diffeomorphisms to enforce hyperbolicity through a cone-field criterion. The Gluing Theorem extends Béguin–Bonatti–Yu’s block gluing to blocks with attractors/repellers and provides a transitivity criterion via a Smale-type graph, enabling realization of various quasi-transverse laminations and embedding of blocks into Anosov flows. The approach yields powerful applications, including realising prescribed boundary bifoliations, embedding blocks in Anosov flows, and constructing atoroidal JSJ pieces for transitive Anosov dynamics, thereby advancing the understanding and flexibility of 3D Anosov flows.

Abstract

We prove a new result allowing to construct Anosov flows in dimension 3 by gluing building blocks. By a building block, we mean a compact 3-manifold with boundary $P$, equipped with a $C^1$ vector field $X$, such that the maximal invariant set $\cap_{t \in \mathbb{R}} X^t (P)$ is a saddle hyperbolic set, and the boundary $\partial P$ is quasi-transverse to $X$, i.e. transverse except for a finite number of periodic orbits contained in $\partial P$. Our gluing theorem is a generalization of a recent result of F. Béguin, C. Bonatti, and B. Yu who only considered the case where the block does not contain attractors nor repellers, and the boundary $\partial P$ is transverse to $X$. The quasi-transverse setting is much more natural. Indeed, our result can be seen as a counterpart of a theorem by Barbot and Fenley which roughly states that every 3-dimensional Anosov flow admits a canonical decomposition into building blocks (with quasi-transverse boundary). We will also show a number of applications of our theorem.

Figures (56)

• Figure 1: A torus quasi-transverse to a vector field $X$ containing two periodic orbits
• Figure 2: A building block $(P,X)$
• Figure 3: Boundary of a building block in a neighborhood of a periodic orbit contained in the boundary
• Figure 4: The boundary $\partial P$ crosses two opposite quadrants of ${\mathcal{O}}$
• Figure 5: An example of quasi-Morse-Smale lamination on the torus

Theorems & Definitions (258)

• Proposition B: Proposition \ref{['prop: transitivity criterion']}
• Definition 1.1.1: Surface quasi-transverse to a vector field
• Definition 1.1.2: Building block
• Remark 1.1.3
• Remark 1.1.4
• Definition 1.1.5: Entrance boundary and exit boundary
• Claim 1.2.1
• proof : Proof of Claim \ref{['claim: boundary quadrant and multipliers']}
• Remark 1.2.3
• proof : Proof of Proposition \ref{['prop: existence boundary lamination']}