RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 151DEFINITION 5. We say that the pair (D,φ) stretches  to ̂B along the pathsand write(D,φ) : Â⊳̂B,if φ is continuous on D∩A and, moreover, there is a compact set W ⊆ D∩A such that,for every path σ ⊆ A with σ ∩ A − l̸= ∅ and σ ∩ A − r ̸= ∅, there is a path γ ⊆ σ ∩ Wwith ψ(γ) ∩ B − l̸= ∅ and φ(γ) ∩ B − r ̸= ∅.Observe that this definition coincides withaccording to Definition 2, for(D ′ ,φ) : Ã⊳˜B,D ′ = D ∩ A ∩ φ −1 (B).As in [60] we introduce now some special subsets of a cell which are crucial for ourapplications.DEFINITION 6. Let ̂M = (M, c M ; X, d 1 ) and ̂N = (N, c N ; X, d 2 ) be tworectangular cells of the same topological space X and let ˜M = (M,M − ) and Ñ =(N,N − ) be the corresponding path-oriented spaces. We say that ̂M is a horizontalslab of ̂N and writêM ⊆ h ̂N,if ˜M ⊆ h Ñ, that is, if M ⊆ N and if every path γ ⊆ M with γ ∩ M − l̸= ∅ andγ ∩ M − r ̸= ∅ is such that γ ∩ N − l̸= ∅ and γ ∩ N − r ̸= ∅.Similarly, we say that ̂M is a vertical slab of ̂N and writêM ⊆ v ̂N,if ˜M ⊆ v Ñ, that is, if M ⊆ N and if every path σ ⊆ N with σ ∩ N − l̸= ∅ andσ ∩ N − r ̸= ∅ contains a sub-path γ ⊆ M such that γ ∩ M − l̸= ∅ and γ ∩ M − r ̸= ∅.REMARK 8. Note that in order to have ̂M ⊆ h ̂N it is equivalent to require thatM ⊆ N and either̂M − l⊆ ̂N − l, ̂M − r ⊆ ̂N − ror̂M − l⊆ ̂N − r , ̂M − r ⊆ ̂N − l.In this manner, our definition of horizontal slab reduces to the one of horizontal slicein [60, Def.1.2]As in Section 3, we can now borrow from [60] and [61] the next definition.