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A note on strange nonchaotic attractors. (English) Zbl 0899.58033

Let \(X=T^1 \times [0,\infty)\). The author studies a class of quasiperiodically forced time-discrete dynamical systems \(T: X \to X\) given by \[ T(\theta , x) = (\theta+\omega, f(x) \cdot g(\theta)), \] where \(\omega \in \mathbb{R} \setminus \mathbb{Q}\), \(f: [0,\infty) \to [0,\infty)\) is bounded \(C^1\) and \(g : T^1 \to [0,\infty)\) is continuous. The author further assumes that \(f(0)=0\) and that \(f\) is increasing and strictly concave. The dynamical system is also assumed to have nonpositive Lyapunov exponents. Under the above assumptions, the author proves the existence of an attractor \(\overline \Gamma\) with the following properties:
1. \(\overline \Gamma\) is the closure of the graph of a function \(x=\phi(\theta)\). It attracts Lebesgue-a.e. starting points in \(T^1 \times \mathbb{R}_+\). The set \(\{ \theta: \phi(\theta) \not= 0 \}\) is meager but has full 1-dimensional Lebesgue measure.
2. The omega-limit of Lebesgue-a.e. point in \(T^1 \times \mathbb{R}_+\) is \(\overline \Gamma\), but for a residual set of points in \(T^1 \times \mathbb{R}_+\) the omega limit is the circle \(\{(\theta,x): x=0\}\) contained in \(\overline \Gamma\).
3. \(\overline \Gamma\) is the topological support of a BRS measure. The corresponding measure-theoretical dynamical system is isomorphic to the forcing rotation.

MSC:

37A99 Ergodic theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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