Piecewise contractions defined by iterated function systems

Date(s) : 03/02/2015
11 h 00 min - 12 h 00 min

We are interested in the asymptotic behavior of piecewise contractions of the interval (PCs). A map f : [0, 1) → [0, 1) is a PC of n intervals if there exists a partition of I = [0, 1) into subintervals I1, . . . , In such that each restriction f|Ii : Ii → I is a Lipschitz-continuous contraction. We consider here an (n − 1)-parameter family C of PCs of n intervals defined by an Iterated Function System {φ1, . . . , φn}. More specifically, let φ1, . . . , φn : [0, 1] → (0, 1) be a given collection of Lipschitz-continuous piecewise contractions. We prove that for Lebesgue almost every 0 = x0 < x1 < · · · < xn−1 < xn = 1, the PC of n intervals f : [0, 1) → [0, 1) defined by x ∈ [xi−1, xi) → φi(x) has at least one and at most n periodic orbits. Besides, the w-limit set of every point x ∈ [0, 1) is a periodic orbit. Joint work with Arnaldo Nogueira and Rafael Rosales. [http://dcm.ffclrp.usp.br/~benito/]




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