Foliations of surfaces and semi-Markovian subsets of subshifts of finite type

Let S be a closed surface of genus g ⩾ 2. In this paper, we consider a space, which we call F, of equivalence classes of measured foliations of S, defined as the quotient of Thurstonʹs measured foliation space where one forgets the transverse measure associated to a measured foliation. We give a presentation, in the sense of symbolic dynamics, of the action of a pseudo-Anosov mapping class of M in the neighborhood of its attracting fixed point in F. The action is semi-Markovian. The elements of the combinatorics associated to the presentation consist in an invariant train track with a marking on its set of vertices and a certain number of elementary moves on it.