RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 149Figure 12: The bold line viewed from a suitably chosen point of perspective representsthe arc in R 3 parameterized by t ↦→ (t,θ 1 (t),θ 2 (t)), for t ∈ [0, 3π].The plot has been performed by Maple software.the time t = 5π/2. Finally, the motion switches again to the counterclockwise senseand the point Q = (−1, 0) is reached at the time t = 3π.Let us set now Z = S 1 and Zl − = {P}, Zr − = {Q} (we intentionally take this choiceto show that the terms “left” and “right” are merely conventional and their order isnot important). According to our definitions, γ is a path in Z with γ ∩ Zl− ̸= ∅ andγ ∩ Zr− ̸= ∅. If we consider now the arcs Ŵ upper := {(x, y) ∈ S 1 : y ≥ 0} andŴ lower := {(x, y) ∈ S 1 : y ≤ 0} which are contained in ¯γ, we see that while Ŵ lower isthe image set of a sub-path of γ, Ŵ upper is not the image of any sub-path of γ.4. Applications to topological cells in finite dimensional spaces, periodic pointsand topological dynamicsIn this section we propose an application of the results in Section 3 to the setting of[26, 56, 57, 59, 60, 61, 82].4.1. DefinitionsLet X be a Hausdorff topological space. We define a (1, N − 1)-rectangular cell of Xas a pair̂N = (N, c N ),where N ⊆ X is a compact set and c N : N → [−1, 1] N ⊆ R N is a homeomorphismof N onto its image [−1, 1] N . Sometimes, it will be convenient to put in evidence theHausdorff topological space X containing a given cell N and the dimension N of thecodomain of the homeomorphism. In such a situation, we’ll writêN = (N, c N ; X, N).