Let 0 < a < 1, 0 ≤ b < 1 and I = [0,1). We call contracted rotation the interval map φa,b : x ∈ I → ax+b mod1. Once a is fixed, we are interested in the dynamics of the one-parameter family φa,b, where b runs on the interval interval [0, 1). Any contracted rotation has a rotation number ρa,b which describes the asymptotic behavior of φa,b. In the first part of the talk, we analyze the numerical relation between the parameters a, b and ρa,b and discuss some applications of the map φa,b. Next, we introduce a generalization of contracted rotations. Let −1 < λ < 1 and f : [0, 1) → R be a piecewise λ-affine contraction, that is, there exist points 0 = c0 < c1 < ··· < cn−1 < cn = 1 and real numbers b1,…,bn such that f(x) = λx + bi for every x ∈ [ci−1,ci). We prove that, for Lebesgue almost every δ ∈ R, the map fδ = f + δ (mod 1) is asymptotically periodic. More precisely, fδ has at most n + 1 periodic orbits and the ω-limit set of every x ∈ [0, 1) is a periodic orbit.
Venue: Beauchef 851, Torre Norte Piso 7, Sala de Seminarios John Von Neumann CMM.
Speaker: PROF. ARNALDO NOGUEIRA
Affiliation: INSTITUT DE MATHÉMATIQUES DE MARSEILLE, FRANCIA
Coordinator: Prof. Michael Schraudner