RENDICONTI DEL SEMINARIO MATEMATICO

144 D. Papini - F. ZanolinFigure 11: A possible illustration of Example 2 in R 3 , where we have denoted by Pthe point h −1 ({0}) and by A the surface h −1 (K l ) of the deformed cone Z. Accordingto our result there exists at least a fixed point z = ψ(z) ∈ W, for any continuous mapψ defined on a closed subset W of Z and with values in Z having the property that anypath σ in Z and joining P to A contains a sub-path γ ⊆ σ ∩ W with ψ(γ) ⊆ Z andψ(γ) joining P to A.(i3 ′ ) there is a closed set W ⊆ D such that, for every continuum σ ⊆ Z withσ∩Zl − ̸= ∅ and σ∩Zr − ̸= ∅, there is a continuum Ŵ ⊆ σ∩W with ψ(Ŵ)∩Zl − ̸=∅ and φ(Ŵ) ∩ Zr − ̸= ∅;there exists at least a fixed point of ψ in D.In the present paper we do not further pursue the research in this direction andconfine ourselves to the study of the stretching condition along the paths. Investigationstoward the fixed point properties for maps satisfying an expansive conditions withrespect to other kind of connected sets will be considered elsewhere.3.2. Further definitions and consequencesAs a next step, we give now some simple (but nevertheless useful) properties about thestretching along the paths condition. Unless otherwise specified, all the spaces involvedare arcwise connected topological spaces. When we consider a triple (Z, Zl − , Zr − ), wealways assume that Zl − and Zr − are nonempty disjoint subsets of Z. First of all, weconsider two further definitions.DEFINITION 3. Let ˜Z = (Z, Z − ) and Ỹ = (Y, Y − ) be two path-orientedspaces with Y a subspace of Z. We say that Ỹ is a horizontal slab of ˜Z and writeif every path γ ⊆ Y with γ ∩ Y −lỸ ⊆ h˜Z,̸= ∅ and γ ∩ Y −r ̸= ∅ is such that γ ∩ Z − l̸= ∅ and