RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 145γ ∩ Z − r ̸= ∅.Similarly, we say that Ỹ is a vertical slab of ˜Z and writeỸ ⊆ v ˜Z,if every path σ ⊆ Z with σ ∩ Zl − ̸= ∅ and σ ∩ Zr − ̸= ∅ contains a sub-path γ ⊆ Ysuch that γ ∩ Yl− ̸= ∅ and γ ∩ Yr − ̸= ∅.REMARK 6. The definition of slabs generalizes the case of rectangles with horizontaland vertical sides parallel to the contracting and expanding directions in theSmale horseshoe (see, for instance, [72, Section 2.3]). In our general setting of atopological space Z oriented by the paths connecting two disjoint subsets Zl− and Zr−and in view of Theorem 6, we consider as horizontal-expanding the “direction” along(Zl − , Zr − ) (of course, in a very vague sense and taking also into account the fact thatin our setting “horizontal” and “vertical” are merely conventional terms). Definition3 generalizes the analogous concepts of “slices” considered in [60] in the setting oforiented two-dimensional cells and recalled in Section 1.2 as well as some possibilitiesconsidered in [61] for N-dimensional cells (namely, the case in which there is a onedimensionalexpansive direction). Note that our definitions are purely topological innature and therefore we do not need (like in [72, Section 2.3]) the slabs to be describedby means of graphs of Lipschitz functions. We refer to Figure 14 as a possible pictureof horizontal and vertical slabs in a simple situation.Having available in the general setting the definition of slabs, we can now borrowfrom [60] and [61] the next definition (compare also to the corresponding definitionin Section 1.2).DEFINITION 4. Let ˜Z = (Z, Z − ), Ỹ = (Y, Y − ) and ˜X = (X, X − ) be threepath-oriented spaces with X, Y, Z subspaces of the same topological space W andX ⊆ Y ∩ Z.We say that Ỹ crosses ˜Z in ˜X and write˜X ∈ {˜Z ⋔ Ỹ},if˜X ⊆ h ˜Z and ˜X ⊆ v Ỹ.REMARK 7. As already remarked in [60] and [61], in our setting, the definitionof ˜X ∈ {˜Z ⋔ Ỹ}, covers very general situations, in particular also when there is no wayto define any kind of transversal intersection. A possible illustration is given in Figure15 of Section 4.LEMMA 4. The following properties hold:(e 1 ) if (D,ψ) : ˜Z⊳Ỹ and (E,φ) : Ỹ⊳˜X, then (F,φ ◦ ψ) : ˜Z⊳˜X for F =D ∩ ψ −1 (E);