RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 143and we also set C − := C l ∪ C r . Theorem 6 implies that the path-oriented space˜C = (C,C − ) has the FPP-γ and therefore Lemma 3 ensures that the same fixed pointproperty holds also with respect to retracts of˜C. The closed unit ball B[0, 1] is a retractof the cylinder C through the continuous map ̺ defined by̺(x) = (x 1 min {1,δ(x)} ,..., x N−1 min {1,δ(x)} , x N ),where√1 − xN2δ(x) = δ(x 1 ,..., x N−1 , x N ) := √.x1 2 + ··· + x2 N−1Hence, the path oriented space ˜B = (B, B − ), with B = B[0, 1], B − = B − l∪ B − r ,B − l= {South pole} and B − r = {North pole} has the FPP-γ. Finally, Lemma 2 impliesthe the FFP-γ holds for every oriented space in which the base space Z is homeomorphicto a closed ball and we select as Z − land Z − r two different points of Z. Thisconcludes the proof.EXAMPLE 2. Consider the coneK = {(x 1 ,..., x k , x k+1 ) : ||(x 1 ,..., x k )|| ≤ x k+1 ≤ 1} ⊆ R k+1and select the point 0 = (0,...,0, 0) ∈ K and the base K l = {(x 1 ,..., x k , 1) :||(x 1 ,..., x k )|| ≤ 1} ⊆ K. Let Z ⊆ E be a compact set which is homeomorphic to K,by a homeomorphism h : Z → K. Define Z − r = h −1 ({0}) and Z − l= h −1 (K l ). Then˜Z has the FPP-γ.Proof. It is possible to obtain our claim by suitably adapting the argument employedin the proof of Example 1. We omit the details.REMARK 5. We observe that one could define a fixed point property (say FPP-Ŵ) for maps satisfying a condition which extends property (H ′ ) to general (oriented)topological spaces. Then, after having obtained from Theorem 7 a result analogous toLemma 2, the following corollary can be proved.COROLLARY 4. Let K ̸= ∅ be a compact convex subset of a normed space.Let Z be a compact topological space which is homeomorphic to [−1, 1] × K, via ahomeomorphism h : Z → [−1, 1] × K. DefineZ − l:= h −1 ({−1} × K), Z − r := h −1 ({1} × K).Then, for every pair (D,ψ), satisfying the following conditions:(i 1 ) D ⊆ Z;(i 2 ) ψ : D → Z is continuous;