RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 127Figure 9: (compare to Figure 6). Now the worm crosses nicely the cheese for two timesand we obtain a complete dynamics on two symbols (as well as periodic points of anyperiod).three regions with corresponding symbols, say 1, 2, 3, we have also that to any periodicsequence in {1, 2, 3} we can find a corresponding periodic point for the map ψ whichfollows (along ψ) the itinerary described by the periodic sequence. Figure 8 illustratesthe case in which ψ is a homeomorphism and the three good intersections are pairwisedisjoint. Actually, our Theorem 3 is more flexible (cf. Corollary 2 below) and it allowsto come to the same conclusion by a careful selection of disjoint subsets of the domainof the map (which is not necessarily a homeomorphism). As an illustration of thisremark, let us consider Figure 9.In this direction, two possible corollaries of Theorem 3 are the following. They correspond,respectively: (a) to the property (H ± ) which holds with respect to thesolutions of system (1) and the two conical shells W(+) and W(−), and (b) to theexample depicted in Figure 9. We refer to [59] for the proof of both the corollaries, aswell as for the proof of a more general result from which they both come.COROLLARY 1. Let ̂R 0 = (R 0 ,R − 0 ) and ̂R 1 = (R 1 ,R − 1), be two orientedrectangles with R 0 ∩ R 1 = ∅ and such thatThen the following conclusions hold:ψ : ̂R i ⊳̂R j , ∀ i, j ∈ {0, 1}.(c 1 ) ψ has a dynamics of coin-tossing type with respect to the pair (R 0 ,R 1 ).;(c 2 ) For every sequence s = (s n ) n , with s n ∈ {0, 1} for each n ≥ 0, there is acontinuum C s ⊆ R s0 satisfyingC s ∩ (R s0 ) + b ̸= ∅, Cs ∩ (R s0 ) + t ̸= ∅