RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 141there exists at least a fixed point of ψ in D.DEFINITION 2. Suppose we have two arcwise connected topological spacesZ, Y and assume that Zl − , Zr − are two nonempty disjoint subsets of Z with Z − =Zl− ∪ Zr − −, as well as Yl, Yr − are two nonempty disjoint subsets of Y, with Y − =Yl− ∪ Yr− . Defining, as above ˜Z = (Z, Z − ) and Ỹ = (Y, Y − ) the correspondingpath-oriented spaces, we say that the pair (D,ψ) stretches ˜Z to Ỹ along the paths andwrite(D,ψ) : ˜Z⊳Ỹ,if the conditions( j 1 ) D ⊆ Z;( j 2 ) ψ : D → Y is continuous;( j 3 ) there is a closed set W ⊆ D such that, for every path σ ⊆ Z with σ ∩ Zl − ̸= ∅and σ ∩ Zr− ̸= ∅, there is a path γ ⊆ σ ∩ W with ψ(γ) ∩ Yl− ̸= ∅ andφ(γ) ∩ Yr − ̸= ∅;hold.Accordingly, we have that ˜Z has the FPP-γ if and only if for every pair (D,ψ) with(D,ψ) : ˜Z⊳˜Z, there is at least a fixed point of ψ in D.REMARK 4. The definition of a map stretching along the paths was introducedin [56, 57] and refined in [59, 60] in the case of two-dimensional oriented cells. OurDefinition 2 above is a generalization of the previous cited one as it reduces to [60]in the situation considered therein. We note that in [60], as well as in the other precedingpapers, the map ψ was allowed to be defined possibly on some larger domains.However, up to a restriction, we can always enter in the case of Definition 2 when weconsider the situation described in [60].In the definition of path-oriented space, as well as in the subsequent stretching condition,the order in which we label the two sets Zl − and Zr − (or Yl− and Yr − ) has no effectat all.We also point out that given two nonempty disjoint sets W l and W r of a topologicalspace W, the condition that there exists a path σ ⊆ W with σ ∩ W l ̸= ∅ andσ ∩ W r ̸= ∅ is equivalent to the existence of a continuous map θ : [0, 1] → W withθ(0) ∈ W l and θ(1) ∈ W r .The choice of the notation ˜Z instead of Ẑ (previously considered in Section 1.2 andnext again in Section 4.1) comes from the fact that, even if all the applications wepresent here are for the ̂[ ·]-sets, nonetheless, the oriented spaces ˜[ ·] are, in principle,more general. Thus we prefer to think to the ̂[ ·]-sets as some particular cases of the˜[ ·]-sets.Finally, we mention that some analogous definitions, previously introduced in the literature(see, for instance, the concept of quadrilateral set given in [36]) also fit with ourdefinition of oriented space.