RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 131tions to bifurcation (and co-bifurcation) theory [20, 21, 22, 62], to the investigation ofthe structure of the solution set for parameter dependent equations [19, 31, 43] and tothe study of nonlinear problems in absence of a priori bounds [7, 10, 44, 46]. Thus, asa byproduct of our proof of Theorem 6 we also provide a new proof of the main fixedpoint result for planar maps in [57], without the need to rely on properties of planetopology.Then, we give some variants of Theorem 6 which are analogous to the differentforms in which the Schauder fixed point theorem is usually presented. In Section 3 weinvestigate an abstract fixed point property for topological spaces which express in amore abstract fashion the content of Theorem 6 and its variants. An analysis of such anew fixed point property allows (like in the case of the classical fixed point property)to prove that it is invariant under homeomorphisms as well as it is preserved undercontinuous retractions. This in turns, permits to obtain some general results in whichwe produce fixed points for maps defined on topological cylinders. By topologicalcylinders we mean sets which are obtained from a cylinder like B[a, R] after a deformationgiven by a homeomorphism. For instance, the following result (see Corollary 3of Section 3.1) is obtained.THEOREM 5. Let K ̸= ∅ be a compact convex subset of a normed space. LetZ be a compact topological space which is homeomorphic to [0, 1] × K, via a homeomorphismh : Z → [0, 1] × K. DefineZ − l:= h −1 ({0} × K), Z − r := h −1 ({1} × K).Suppose that ψ : Z ⊇ D ψ → Z is a map which is continuous on a set D ⊆ D ψ andassume the following property is satisfied:there is a closed set W ⊆ D such that for every path σ ⊆ Z with σ ∩ Zl− ̸= ∅and σ ∩ Zr − ̸= ∅, there is a sub-path γ ⊆ σ ∩ W with φ(γ) ∩ Zl − ̸= ∅,φ(γ) ∩ Zr − ̸= ∅.Then there exists a fixed point ˜z of ψ with ˜z ∈ D (actually, ˜z ∈ W).The possibility of studying topological cylinders (instead of topological rectangleslike in [59, 60]) open the way toward an extension of the results about orientedrectangles presented in Section 1.3 to higher dimensional objects possessing a privilegeddirection. Thus we conclude the paper with a list of possible applications tomaps which stretch the paths along a direction in a (1, N − 1)-rectangular cell (seeSection 4.1 for the corresponding definition).1.5. NotationThroughout the paper, the following notation is used. Let Z be a topological space andlet A ⊆ B ⊆ Z. By cl B A and int B A we mean, respectively, the closure and the interiorof A relatively to B (that is, as a subset of the topological space B with the topology