Chaotic behaviour of the map x ↦ ω(x, f)

Emma D’Aniello 1  and Timothy Steele 2
  • 1 Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, Viale Lincoln 5, 81100, Caserta, Italy
  • 2 Department of Mathematics, Weber State University, 1702, University Circle, Ogden, UT, 84408-2501, USA

Abstract

Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and some forms of chaos are investigated.

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