A new combinatorial invariant caracterizing Anosov flows on 3-manifolds

TL;DR

This paper introduces a new combinatorial invariant based on Markovian families of rectangles in the bifoliated plane, providing a classification method for transitive Anosov flows on 3-manifolds up to orbital equivalence.

Contribution

It generalizes Markov partitions to Markovian families and associates geometric types to classify Anosov flows via combinatorial objects.

Findings

Existence of infinitely many Markovian families for any transitive Anosov flow.

Each geometric type fully describes the flow up to certain surgeries.

Classification of flows using finite collections of combinatorial objects.

Abstract

In this paper, we describe a new approach to the problem of classification of transitive Anosov flows on 3-manifolds up to orbital equivalence. More specifically, generalizing the notion of Markov partition, we introduce the notion of Markovian family of rectangles in the bifoliated plane of an Anosov flow. We show that any transitive Anosov flow admits infinitely many Markovian families, each one of which can be canonically associated to a finite collection of combinatorial objects, called geometric types. We prove that any such geometric type describes completely the flow up to Dehn-Goodman-Fried surgeries on a finite set of periodic orbits of the flow. As a corollary of the previous result, we show that any Markovian family can be canonically associated to a finite collection of combinatorial objects, called geometric types with cycles, each describing the flow up to orbital equivalence.