RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 153Figure 15: In R 3 , the (1, 2)-rectangular cell ̂N (the cheese shaped set) is crossed bythe (1, 2)-rectangular cell ̂M (the snake-like set). Among the four intersections of Mwith N, the first two (counting from the left and painted by a darker color) belong to{̂N ⋔ ̂M}.then the following conclusion holds:• ψ has a chaotic dynamics of coin-tossing type on k symbols (with respect to thesets K i := D ∩ M i ).• ψ has a fixed point in each set K i := D ∩ M i 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 ∗ .REMARK 9. The two conclusions in Theorem 12 corresponds to (a 1 ) and (a 3 )of Theorem 11. We could derive from (a 2 ) also a conclusion about the existence of acontinuum of initial points which generate any (fixed) forward itinerary and thus obtainan extension of the conclusion (b 2 ) of Theorem 4. This one as well as some relatedtopics, which require a more careful treatment, will be discussed elsewhere.As shown by this example, from Theorem 11 and the definitions of stretching,slabs and crossings adapted to the case of (1, N − 1)-rectangular cells, we have now