Open book decompositions of a suspension Anosov flow and affine structures.
Despite the good comprehension that we have nowadays about the asymp- totic dynamical behaviour of a general Anosov flow, classification of the different orbital equivalent classes rest a major subject nowadays. In the particular case of dimension three, a lot of different examples of Anosov flows can be constructed using surgery methods, which shows that the set of different equivalence classes is not at all simple to describe.
If we restrict to the special subfamily of transitive 3-dimensional Anosov flows, each flow has (many) associated open book decompositions of the 3- manifold with pseudo-Anosov monodromy. Since pseudo-Anosov homeomor- phisms can be classified by terms of a combinatorial invariant, there is a hope to produce combinatorial invariants of the orbital equivalence class of these Anosov flows in terms of those available for pseudo-Anosov. The problem to solve is : Given two open book decompositions with pseudo-Anosov monodromy, how to determine if both correspond to the same flow?
We study a simpler question related to the previous one: Given an open book decomposition with pseudo-Anosov monodromy, can we determine whether or not the corresponding Anosov flow is a suspension Anosov flow ? In this talk we will provide a criterion for this question based on the existence of some affine incomplete structures associated with the pseudo-Anosov monodromy. In turn, we can provide a natural bijection between the set of genus one Birkhoff sections of a suspension Anosov flow and these affine structures.