Skewed Anosov Flows in 3-Manifolds

• Skewed Anosov flows are a class of Anosov flows defined on closed 3-manifolds with a uniquely twisted orbit space structure characterized by diagonal or anti-diagonal strips.
• They are constructed through Dehn-Goodman-Fried surgery on simple closed orbits, with invariant foliations and linking numbers playing a key role in determining their dynamic properties.
• Positively skewed flows are orbit equivalent to Reeb-Anosov flows, connecting hyperbolic dynamics, contact topology, and bi-contact geometry to yield deeper insights into low-dimensional topology.

A skewed Anosov flow is a class of Anosov flow on a closed, oriented 3-manifold, characterized by a geometrically distinct orbit space structure, significant connections to contact and foliation theory, and deep ties to the theory of Reeb flows and surgery constructions. In dimension three, all positively skewed $\mathbb{R}$-covered Anosov flows are orbit equivalent to Reeb-Anosov flows, linking the dynamics of hyperbolic flows, contact topology, and surgery techniques in low-dimensional topology (Marty, 2023, Bonatti et al., 2020, Salmoiraghi, 2021).

1. Anosov Flows, Orbit Spaces, and the Skewed Case

An Anosov flow $\varphi^t$ on a closed 3-manifold $M$ is a $C^1$ flow such that the tangent bundle splits as $TM=E^s\oplus\langle X\rangle\oplus E^u$, with $X$ the generator and $E^s,E^u$ invariant under $D\varphi^t$, and there exist $A,B>0$ s.t. for all $v\in E^s$, $\varphi^t$0, $\varphi^t$1 (and similarly for $\varphi^t$2 with $\varphi^t$3). In dimension 3 the distributions $\varphi^t$4 are one-dimensional and integrate to strong stable and unstable foliations $\varphi^t$5.

Lifting $\varphi^t$6 to the universal cover and passing to the orbit space $\varphi^t$7, the (weak) stable and unstable foliations project to two transverse one-dimensional foliations $\varphi^t$8. The flow is called $\varphi^t$9-covered if the leaf spaces of $M$0 and $M$1 are homeomorphic to $M$2. The possible topological types of $M$3 for $M$4-covered flows are:

• A product $M$5 with horizontal and vertical lines (suspension case).
• The diagonal strip $M$6 (positively skewed).
• The anti-diagonal strip $M$7 (negatively skewed) (Marty, 2023, Bonatti et al., 2020).

The skewed (twisted) cases derive their name from the embedding of the foliations: rather than globally parallel, the foliations are skewed to give a strip with boundary $M$8 or $M$9. For oriented $C^1$0, $C^1$1 (positively skewed) and $C^1$2 (negatively skewed) are distinct.

2. Surgery Constructions and Bi-foliated Planes

Skewed Anosov flows arise naturally via Dehn-Goodman-Fried surgery on simple closed orbits of canonical Anosov flows, such as suspension or geodesic flows. In this context, given an Anosov flow $C^1$3 and a periodic orbit $C^1$4, a surgery of characteristic number $C^1$5 is performed by modifying $C^1$6 along a neighborhood of $C^1$7 and regluing tori so that the meridian is sent to $C^1$8. If all characteristic numbers in a multi-surgery have the same sign, Fenley's theorem establishes that the resulting flow is $C^1$9-covered and twisted: positive sign yields positively skewed, negative sign negatively skewed (Bonatti et al., 2020).

The bi-foliated plane $TM=E^s\oplus\langle X\rangle\oplus E^u$0 encodes the normal structure to the flow. In the $TM=E^s\oplus\langle X\rangle\oplus E^u$1-covered skewed case, the plane is the diagonal (or anti-diagonal) strip, and both transverse foliations are genuine lines without branching. This structural result provides a toolkit for the construction and recognition of skewed $TM=E^s\oplus\langle X\rangle\oplus E^u$2-covered flows via signs in surgery data, holonomy formulas, and the completeness of certain quadrants in the bi-foliated plane (Bonatti et al., 2020).

3. Orbit Equivalence to Reeb-Anosov Flows

A Reeb-Anosov flow is a Reeb flow of a contact form $TM=E^s\oplus\langle X\rangle\oplus E^u$3 (i.e., $TM=E^s\oplus\langle X\rangle\oplus E^u$4 maximally non-integrable and $TM=E^s\oplus\langle X\rangle\oplus E^u$5 everywhere) which is also Anosov. The central theorem states that any $TM=E^s\oplus\langle X\rangle\oplus E^u$6-covered positively skewed Anosov flow $TM=E^s\oplus\langle X\rangle\oplus E^u$7 on an oriented closed 3-manifold $TM=E^s\oplus\langle X\rangle\oplus E^u$8 is orbit equivalent to a Reeb-Anosov flow (i.e., there is a homeomorphism $TM=E^s\oplus\langle X\rangle\oplus E^u$9 sending orbits of $X$0 bijectively, orientation-preservingly, to those of a Reeb flow $X$1) (Marty, 2023).

This statement settles a conjecture of Barbot-Barthelmè and fully characterizes the positively skewed $X$2-covered class in dimension three: all such flows are “of Reeb type”, and conversely any Reeb-Anosov flow is necessarily $X$3-covered and skewed.

The orbit equivalence is constructed using regularization of invariant measures and smoothing arguments on charts, combined with the McDuff–Prasad criterion: a smooth flow on $X$4 is a reparametrization of a Reeb flow if and only if it preserves a smooth volume $X$5 such that the contraction $X$6 is exact, and all linking numbers with null-homologous invariant probabilities are positive.

4. Linking Numbers, Birkhoff Sections, and Invariant Forms

The existence of an invariant contact form, or of a Birkhoff section with prescribed boundary, is intimately connected to linking numbers between invariant signed measures. For an oriented embedded surface $X$7 transverse to $X$8 with boundary a union of periodic orbits, $X$9 is called a Birkhoff section if every orbit segment of length $E^s,E^u$0 meets $E^s,E^u$1. Fried proved the existence of Birkhoff sections for transitive Anosov flows; recent results provide linking number criteria for when a null-homologous algebraic multi-orbit bounds such a section (Marty, 2023).

The extended linking pairing $E^s,E^u$2 on signed invariant measures is symmetric, bilinear, and continuous. A signed measure is Reeb-like if it is null-homologous and has positive linking with all other such invariant null-homologous probability measures. The existence of an invariant contact form or Birkhoff section reduces to satisfying positivity of this extending linking pairing.

The McDuff–Prasad criterion precise the conditions under which a smooth flow can be realized as a reparametrization of a Reeb flow: the necessary and sufficient condition is the existence of an invariant smooth positive volume $E^s,E^u$3 such that $E^s,E^u$4 is exact and linking with every null-homologous invariant probability is positive (Marty, 2023).

5. Bi-contact Geometry and Equivalence of Surgery Techniques

The construction and manipulation of skewed Anosov flows via Dehn-Goodman-Fried surgery can be re-expressed in the framework of bi-contact geometry. A bi-contact structure is a pair of transverse, oppositely oriented contact structures on $E^s,E^u$5, whose intersection contains an Anosov flow with orientable weak foliations. The Foulon–Hasselblatt Legendrian surgery and the Goodman surgery are shown to be orbit equivalent under suitable conditions, clarifying when new Anosov (resp. Reeb-Anosov/contact Anosov) flows arise from surgeries (Salmoiraghi, 2021).

Explicit local models demonstrate that Legendrian–transverse surgery in the bi-contact category produces Anosov flows if and only if the surgery slope parameter $E^s,E^u$6 satisfies $E^s,E^u$7; equivalently, for geodesic flows on $E^s,E^u$8, every $E^s,E^u$9-Dehn surgery along a simple closed geodesic produces a contact Anosov flow, and any positively skewed $D\varphi^t$0-covered Anosov flow obtained in this manner is orbit equivalent to a positive contact Anosov flow (Salmoiraghi, 2021).

6. Open Book Decompositions and Applications

A major structural consequence is that positively skewed $D\varphi^t$1-covered Anosov flows on 3-manifolds admit open book decompositions with a single boundary component, arising from Birkhoff sections with one binding orbit. Here, the flow is dynamically supported by the pages of the open book, and the binding orbits serve as boundaries. The existence of such decompositions for Reeb-Anosov (hence positively skewed) flows ties the dynamics to topological invariants and contact structures (Marty, 2023).

This structure has broad implications: it refines the classification of Anosov flows into a tetrachotomy (positively twisted, flat, negatively twisted, other), and enables concrete identifications and orbital equivalences in geometric topology and dynamical systems. It also elucidates the interrelation of hyperbolic dynamics, contact topology, and surgery theory in the setting of 3-manifolds.

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Key References:

• "Skewed Anosov flows are orbit equivalent to Reeb-Anosov flows in dimension 3" (Marty, 2023)
• "Anosov flows on $D\varphi^t$2-manifolds: the surgeries and the foliations" (Bonatti et al., 2020)
• "Surgery on Anosov flows using bi-contact geometry" (Salmoiraghi, 2021)