A new combinatorial invariant characterizing Anosov flows on 3‐manifolds

Source abstract

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. To every transitive Anosov flow on is associated a bifoliated plane endowed with an action of . Thanks to a theorem of Barbot, the previous action characterizes up to orbital equivalence. The goal of this paper is to classify the above class of actions arising from transitive Anosov flows using Markovian families , introduced here as a group‐action analogue of Markov partitions. More specifically, we prove that every transitive Anosov flow admits infinitely many Markovian families; given a Markovian family of , say , the number of orbits of rectangles of and their pattern of intersection can be encoded by a finite combinatorial object, called a geometric type, which describes completely up to Dehn–Goodman–Fried surgeries on a finite set of periodic orbits of ; equipping a geometric type of with additional combinatorial data, referred to as cycles , produces a finite combinatorial invariant that completely characterizes up to orbital equivalence.