RENDICONTI DEL SEMINARIO MATEMATICO

124 D. Papini - F. ZanolinFigure 8: (taken from [60]). Example of oriented cells ˜R (white) and ψ(˜R) (light color)with crossings into three slabs (darker color). The [ ·] − -sets are indicated with a boldline. Among the five cells which are the connected components of the intersectionψ(R) ∩ R, only the three painted with darker color are suitable to play the role of theM’s for the application of Theorem 2.Now, given three oriented rectangles (cells) in X, which are denoted by  = (A,A − ),̂B = (B,B − ) and ̂M = (M,M − ), we say that ̂B crosses  in ̂M and writêM ∈ { ⋔ ̂B},if̂M ⊆ h  and ̂M ⊆ v ̂B.The symbol ⋔ is borrowed from the case of transversal intersections, however we pointout that in our situation (although confined to sets which are two-dimensional in nature)we don’t need any smoothness assumption. In fact, our setting is that of topologicalspaces. From the above definitions and by Theorem 1 the following result easily follows.THEOREM 2. Let  = (A,A − ) and ̂B = (B,B − ) be oriented cells in X. If(D,K,ψ) : Â⊳̂B and there is an oriented cell ̂M such that ̂M ∈ { ⋔ ̂B}, thenthere exists w ∈ K ∩ M such that ψ(w) = w.A situation like that depicted in Figure 8 in which we have more than one goodintersection between the domain and the image of a homeomorphism is typical of thehorseshoe maps and thus, as a next step, we can look for the existence of a completedynamics on m symbols, where m ≥ 2 is the number of the crossings. With thisrespect, we have to recall that a very general topological theory has been developed in