RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 133is a map (not necessarily continuous on its whole domain D φ even if, in the sequel, weassume the continuity of φ on some relevant subset D of D φ ).Let D ⊆ D φ be a given set (in our applications we’ll usually take D closed, forinstance, D = W of Theorem 6 below, but such an assumption for the moment is notrequired). We are looking for fixed points of φ belonging to D, i.e., we want to provethe existence of a pair ˜z = (˜t, ˜x) ∈ D which solves the equation{ t = φ1 (t, x)Our first result is the following.x = φ 2 (t, x).THEOREM 6. Let B[a, R] := [−a, a] × B[0, R] and defineB l := {(−a, x) : ‖x‖ ≤ R}, B r := {(a, x) : ‖x‖ ≤ R}the left and the right bases of the cylinder B[a, R]. Assume thatφ is compact on D ∩ B[a, R]and there is a closed subset W ⊆ D ∩ B[a, R] such that the assumption(H) for every path σ ⊆ B[a, R] with σ ∩ B l ̸= ∅ and σ ∩ B r ̸= ∅, there is asub-path γ ⊆ σ ∩W with φ(γ) ⊆ B[a, R] and φ(γ)∩B l ̸= ∅, φ(γ)∩B r ̸= ∅,holds. Then there exists ˜z = (˜t, ˜x) ∈ W ⊆ D, with φ(˜z) = ˜z.Proof. First of all we observe that, as a consequence of Dugundji Extension Theoremand Mazur’s Lemma (see, e.g., [64, p.22] in the case of Banach spaces or [16, Th.2.5,p.56] for a general situation), there exists a compact operator ˜φ defined on R× X whichextends φ restricted to W, i.e.Consider also the projectionand define the compact operator˜φ : R × X → R × X, ˜φ| W = φ| W .P R : X → B[0, R], P R (x) := x min{1, R ‖x‖ −1 }ψ = (ψ 1 ,ψ 2 ), ψ 1 (t, x) := ˜φ 1 (t, x), ψ 2 (t, x) := P R ( ˜φ 2 (t, x))Note that if ¯z = (¯t, ¯x) is a fixed point of ψ with(6) ¯z ∈ W and φ 2 (¯t, ¯x) ∈ B[0, R],then ¯t = ψ 1 (¯z) = ˜φ 1 (¯z) = φ 1 (¯z) and ¯x = ψ 2 (¯z) = P R ( ˜φ 2 (¯z)) = P R (φ 2 (¯z)) =φ 2 (¯t, ¯x), so that ¯z ∈ W is a fixed point of φ.