ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 58 Zgliczynski, Piotr, Jagiellonian University, Cracow, Poland Sharkovskii theorem for multidimensional perturbations of 1-dim maps Let us denote by ◁ the Sharkovskii ordering of positive integers. Let f : ℝ → ℝ be a continuous map, let V = ℝ ⊕ W be Banach space, and let F : [0, 1] × V → V be a continuous and compact map, such that F_{0}(x, w) = (f(x), 0), where x ∈ ℝ, w ∈ W. Suppose f has a point of period n. Then for every r > 0 there exists λ_{0} > 0 such that for λ ≤ λ_{0} holds: if n ◁ m, m ≠ n then F_{λ} has a periodic point of period m in the set ℝ ⊕ B_{W} (r), where B_{W} (r) is an open ball in W of radius r. Reference: P. Zgliczynski, Sharkovskii theorem for multidimensional perturbations of 1-dim maps, Erg. Th. & Dyn. Sys. , accepted