RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 135is open and connected. Hence, it is arcwise connected as well. Therefore, there is acontinuous map θ : [0, 1] → U ε with θ(0) ∈ B l and θ(1) ∈ B r and, without lossof generality (i.e., possibly cutting off some points of the interval and changing theparameter for the curve) we can also assume thatθ 1 (s) := p 1 (θ(s)) ∈ [−a, a], ∀ s ∈ [0, 1].Next, we define the new curve ζ(s) = (ζ 1 (s),ζ 2 (s)), withζ 1 (s) := p 1 (θ(s)) = θ 1 (s),ζ 2 (s) := P R (p 2 (θ(s))) = P R (θ 2 (s))and observe that ζ(·) satisfies the following properties:(I 1 ) ζ(s) ∈ V ε ∩ B[a, R], ∀ s ∈ [0, 1] ;(I 2 ) ζ(0) ∈ B l and ζ(1) ∈ B r ;where we have setV ε :=N⋃]t i − ε, t i + ε[ ×B(x i , 2ε).i=1To check (I 1 ), let us set x := ζ 2 (s) and assume that ‖x‖ > R as well as x ∈ B(x i ,ε),for some i. Then,Rx∥‖x‖ − x i∥ = ‖Rx − ‖x‖ x i‖/‖x‖≤ R‖x‖ ‖x − x i‖ + (‖x‖ − R ) ‖x i‖‖x‖ < 2ε.The proofs of all the remaining cases for the verification of (I 1 ) are obvious.From (I 1 ) and (I 2 ), it follows that the path σ := [ζ] is contained in the cylinder B[a, R]and it has a nonempty intersection with the left and the right bases of B[a, R]. Then,by hypothesis (H), we know that there exists a sub-path γ of σ, such that γ ⊆ W withφ(γ) ⊆ B[a, R] and φ(γ)∩B l ̸= ∅, φ(γ)∩B r ̸= ∅. Let ξ = (ξ 1 ,ξ 2 ) : [0, 1] → R×Xbe a continuous map such that [ξ] = γ. By the above assumptions, we have that(J 1 ) ξ(s) ∈ V ε ∩ W, ∀ s ∈ [0, 1] ;(J 2 ) φ(ξ(0)) ∈ B l and φ(ξ(1)) ∈ B r ;(J 3 ) φ(ξ(s)) ∈ B[a, R], ∀ s ∈ [0, 1] ;