Markovian families for pseudo-Anosov flows

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 $\Phi$ (with no 1-prongs) on $M^3$ is associated a bifoliated plane $\mathcal{P}$ endowed with an action of $\pi_1(M)$. It is known that the previous action characterizes $\Phi$ 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 $\Phi$, 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 $\Phi$ up to Dehn-Goodman-Fried surgeries on a specific finite set $\Gamma$ of periodic orbits of $\Phi$ 2) our previous choices of surgeries on $\Gamma$ 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 $\Phi$ 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.