A note on strange nonchaotic attractors

نویسنده

  • G. Keller
چکیده

For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ, x) ∈ T1 × R+ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ with the following properties: 1. Γ is the closure of the graph of a function x = φ(θ). It attracts Lebesgue-a.e. starting point in T1 × R+. The set {θ : φ(θ) 6= 0} is meager but has full 1-dimensional Lebesgue measure. 2. The omega-limit of Lebesgue-a.e. point in T1 × R+ is Γ , but for a residual set of points in T1 × R+ the omega limit is the circle {(θ, x) : x = 0} contained in Γ . 3. Γ is the topological support of a BRS measure. The corresponding measure theoretical dynamical system is isomorphic to the forcing rotation. Let X = T1 × [0,∞). We study the dynamical system T : X → X, T (θ, x) = (θ + ω, f(x) · g(θ)) where ω ∈ R \Q, f : [0,∞)→ [0,∞) is bounded C1 and g : T1 → [0,∞) is continuous. We assume furthermore that f(0) = 0 and that f is increasing and strictly concave (i.e. 0 < f ′(x)↘). Define σ := f ′(0) · exp ( \ log g(θ) dθ ) . As g is bounded, the integral in this definition is always well defined, although it may be equal to −∞ in which case it is natural to set σ := 0. (This happens in particular, if g(θ) = 0 for a set of θ’s of positive Lebesgue measure.) Finally, if no ambiguity can arise, we use the notation (θn, xn) = T(θ, x). With this notation we define the vertical Lyapunov exponent at (θ, x) as λ(θ, x) = limn→∞(1/n) log ∂xn/∂x if this limit exists. By λ(θ, x) we denote the corresponding limit superior. In order to make the dependence 1991 Mathematics Subject Classification: 58F11–13. Work partially supported by the DFG.

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تاریخ انتشار 2007