RENDICONTI DEL SEMINARIO MATEMATICO

126 D. Papini - F. ZanolinFrom Theorem 3 several corollaries can be obtained. Now we just recall a fewof them which are taken from [59] and [60]. Due to space limitation, we don’t givehere other applications to ODEs. We just mention the recent thesis by Covolan [12]which contains a detailed description of the results in [33] and those in [59, 60] andwhere it is shown that our theorem, when applied to the search of fixed points forthe iterates of a two-dimensional map, may add some useful information (about theexistence of periodic points) to the conclusions obtained in some recent articles (like,e.g., [30, 75, 76, 77]), where the theory of topological horseshoes was applied to provethe existence of a chaotic dynamics in various different models.THEOREM 4. Suppose that  = (A,A − ) and ̂B = (B,B − ) are oriented cellsin X. If (D,K,ψ) : Â⊳̂B and there are k ≥ 2 oriented cells ̂M 1 ..., ̂M k such thatwitĥM i ∈ { ⋔ ̂B}, for i = 1,...,k,M i ∩ M j ∩ K = ∅, for all i ̸= j, with i, j ∈ {1,...,k},then the following conclusion holds:(b 1 ) ψ has a chaotic dynamics of coin-tossing type on k symbols (with respect to thesets W i = K i = K ∩ M i );(b 2 ) For each one-sided infinite sequence s = (s 0 , s 1 ,...,s n ,...) ∈ {1,...,k} Nthere is a continuum C s ⊆ K s0 withC s ∩ (M s0 ) + l̸= ∅, and C s ∩ (M s0 ) + r ̸= ∅,such that for each point w ∈ C s , the sequencez j+1 = ψ(z j ), z 0 = w, for j = 0, 1,...,n,...satisfiesz j ∈ K s j, ∀ j = 0, 1,...,n,... ;(b 3 ) ψ has a fixed point in each set K i := M i ∩ K and, for each finite sequence(s 0 , s 1 ,...,s m ) ∈ {1,...,k} m+1 , with m ≥ 1, there is at least one point z ∗ ∈K s0 such that the positionz j+1 = ψ(z j ), z 0 = z ∗ , for j = 0, 1,...,mdefines a sequence of points withz j ∈ K s j, ∀ j = 0, 1,...,m and z m+1 = z ∗ .As a comment to this result, we look again at Figure 8 and observe that, besideshaving a coin-tossing dynamics on three symbols, we have also the existence of fixedpoints in each of the three darker regions and, moreover, once we have labelled these