Markovian families for pseudo-Anosov flows

TL;DR

The paper develops a finite, combinatorial framework to classify pseudo-Anosov flows on 3-manifolds up to orbital equivalence by translating flow dynamics into actions on a bifoliated plane $\mathcal{P}$ endowed with stable/unstable foliations. Central to the approach are Markovian families of rectangles and the associated geometric types, which encode the first return dynamics; cycles then track sequences of surgeries on a finite set of periodic orbits. The authors prove that geometric types (and their augmented variants with cycles) determine the original flow up to a finite Dehn-Goodman-Fried surgery on a prescribed finite set of orbits, and they extend the framework to strong Markovian actions on the plane. They also introduce the bifoliated plane up to surgeries $\overline{\mathcal{P}}$ as a robust invariant to compare flows and to realize surgeries in a controlled, finite combinatorial manner. Overall, the work provides a structured, plane-based classification paradigm that parallels Markov partitions for group actions and connects local surgery data to global orbital equivalence, with broad implications for understanding 3-manifold dynamics and associated fundamental-group actions.

Abstract

Generalizing the classification approach described for transitive Anosov flows in dimension 3 in a previous preprint of the author, in this paper we describe a method for classifying (not necessarily transitive) pseudo-Anosov flows on 3-manifolds up to orbital equivalence. To every pseudo-Anosov flow $Φ$ (with no 1-prongs) on $M^3$ is associated a bifoliated plane $\mathcal{P}$ endowed with an action of $π_1(M)$. It is known that the previous action characterizes $Φ$ up to orbital equivalence and admits infinitely many Markovian families (i.e. collections of rectangles in $\mathcal{P}$ generalizing the notion of Markov partition for group actions on the plane). Our goal in this paper consists in showing that : 1) if $\mathcal{R}$ is a Markovian family of $Φ$, the number of orbits of rectangles of $\mathcal{R}$ and their pattern of intersection can be encoded by a finite combinatorial object, called a geometric type, which describes completely $Φ$ up to Dehn-Goodman-Fried surgeries on a specific finite set $Γ$ of periodic orbits of $Φ$ 2) our previous choices of surgeries on $Γ$ can be read as sequences of rectangles in $\mathcal{R}$ and can be encoded by finite combinatorial objects, called cycles 3) a geometric type with cycles of $\mathcal{R}$ describes the original flow $Φ$ up to orbital equivalence Several of the above results will be stated and proven in a slightly more general setting involving strong Markovian actions on the plane. Finally, due to the lack of bibliographic references on pseudo-Anosov flows in dimension 3, in the first part of the paper we provide an introduction to pseudo-Anosov flow theory containing several useful results for our classification approach together with their proofs.

Figures (43)

• Figure 2.1: On the right, the foliations $\mathcal{F}^1_h,\mathcal{F}^1_v$. On the left, $\mathcal{F}^3_h,\mathcal{F}^3_v$
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• Figure 2.4:
• Figure 2.5: The two points above stay close in both the future and the past
• Figure 2.6:

Theorems & Definitions (301)

• Definition
• Definition : Informal
• Definition : Informal
• Definition : Informal
• Definition 2.1
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Definition 2.6
• Definition 2.7