RENDICONTI DEL SEMINARIO MATEMATICO

146 D. Papini - F. Zanolin(e 2 ) if φ : Z → Y is a homeomorphism such that φ(Zl − ) = Yl−(or φ(Zl − ) = Yr− and φ(Zr −) = Y l − ), then (Z,φ) : ˜Z⊳Ỹ ;and φ(Z − r ) = Y −r(e 3 ) if (D,ψ) : ˜Z⊳Ỹ, then (D ∩ X ∩ ψ −1 (W),ψ) : ˜X⊳ ˜W, for every ˜X ⊆ h ˜Zand every ˜W ⊆ v Ỹ ;(e 4 ) if (D,ψ) : ˜Z⊳Ỹ, then ψ has a fixed point in D∩ X, for every ˜X ∈ {˜Z ⋔ Ỹ},having the FPP-γ .Proof. The above properties follow immediately by the corresponding definitions.Now we are in position to consider a sequence of spaces and maps and obtain aresult which is in line with [34] and [73] and extend to a general setting some results[59, Theorem 2.2], [60, Theorem 4.2] previously obtained in the two-dimensional setting.For simplicity, we confine ourselves to the framework of compact metric spaces.This simplifies somehow our proofs. We point out, however, that some of the propertiesexposed in the next Theorem 11 would be still true in some more general situations.THEOREM 11. Suppose that there is a (double) sequence of path-oriented spaces(˜X k ) k∈Z = ((X k , X − k )) k∈Z ,where, for each k ∈ Z, X k is a compact and arcwise connected metric space. Denoteby (Xk − ) l and (Xk − ) r the two sides of Xk − . Assume that there is a sequence of maps((D k ,ψ k )) k∈Z , such thatThen the following conclusions hold:(D k ,ψ k ) : ˜X k ⊳˜X k+1 , ∀ k ∈ Z.(a 1 ) There is a sequence (w k ) k∈Z with w k ∈ D k and ψ k (w k ) = w k+1 for all k ∈ Z;(a 2 ) For each j ∈ Z there is a compact set C j ⊆ D j such that for each w ∈ C jthere exists a sequence (y l ) l≥ j , with y l ∈ D l and y j = w, y l+1 = ψ l (y l ) foreach l ≥ j.The compact set C j satisfies the following separation property:C j ∩σ ̸= ∅, for each path σ ⊆ X j with σ ∩(X − k ) l ̸= ∅ and σ ∩(X − k ) r ̸= ∅;(a 3 ) If there are integers h, k with h < k such that ˜X h = ˜X k and ˜X h possessesthe FPP-γ, then there is a finite sequence (z i ) h≤i≤k , with z i ∈ D i and ψ i (z i ) =z i+1 for each i = h,...,k − 1, such that z h = z k .Proof. As in [59, Theorem 2.2] we prove the three properties in the reverse order.First of all we observe that, for each i ∈ Z, there is a closed (and hence compact)set W i ⊆ D i ⊆ X i such that, for every path σ ⊆ X i with σ ∩ (X i ) − l̸= ∅ andσ ∩ (X i ) − r ̸= ∅, there is a sub-path γ ⊆ σ ∩ W i with ψ i (γ) ∩ (X i+1 ) − l̸= ∅ andψ i (γ) ∩ (X i+1 ) − r ̸= ∅.