A note on strange nonchaotic attractors

Gerhard Keller¹

¹

Fundamenta Mathematicae, Tome 151 (1996) no. 2, pp. 139-148.

Résumé

For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ ${\mathbb T}^1 × ℝ_+$ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̅ with the following properties:  1. Γ̅ is the closure of the graph of a function x = ϕ(θ). It attracts Lebesgue-a.e. starting point in ${\mathbb T}^1 ×ℝ_+$. The set {θ:ϕ(θ) ≠ 0} is meager but has full 1-dimensional Lebesgue measure.   2. The omega-limit of Lebesgue-a.e point in ${\mathbb T}^1 × ℝ_+$ is $Γ̅$, but for a residual set of points in ${\mathbb T}^1 × ℝ_+$ the omega limit is the circle {(θ,x):x = 0} contained in Γ̅.   3. Γ̅ is the topological support of a BRS measure. The corresponding measure theoretical dynamical system is isomorphic to the forcing rotation.

Détail

DOI : 10.4064/fm-151-2-139-148