Entropy Theory for Cross Sections

Suppose that M is a smooth manifold and Tt : M → M is a flow, i.e. a one-parameter group of diffeomorphisms. One way of studying this flow is by looking at a cross-section—that is to say, a codimension 1 submanifold N ⊂ M that is transverse to the direction of the flow. The transversality condition implies that each orbit intersects N in a discrete set. In other words, for each point x of M , the set of visit times, O(x) = {t ∈ R|Tt(x) ∈ N} is discrete. The codimension 1 condition (together with the transversality condition) implies that the set of points whose orbit intersects N contains an open set. Since this set is clearly invariant under the flow, mild irreducibility conditions, such as topological transitivity, imply that almost every orbit (in the sense of Baire category) intersects N . Let us further assume that there is a smooth volume form on M of total measure 1 that is preserved by Tt, and that Tt is ergodic with respect to this measure. In this case, one sees that the negative orbit of N , i.e. the set