The paper studies global attractors of so-called pinched skew products (a class of skew products over irrational rotations of the circle which contains systems with strange nonchaotic attractors). A pinched skew product is a map \(T:X\to X\), \(X=\mathbb{T}^1\times\mathbb{R}\), of the form \[ T(x,y)=(x+\omega ,a(x)+b(x)f(x,y)), \] where \(\omega\in\mathbb{R}\setminus\mathbb{Q}\), \(a\), \(b\) and \(f\) are continuous functions, \(f\) is continuously differentiable and bounded, and \(b(\widehat x)=0\) for at least one \(\widehat x\in\mathbb{T}^1\). The main result of the paper is the following
Theorem. Let \(T\) be a pinched skew product. Then there exist lower and upper semi-continuous functions \(\psi , \phi :\mathbb{T}^1\to\mathbb{R}\) such that the global attractor of \(T\), \(\mathcal A\), is given by \[ {\mathcal A}=\{ (x,y)\in X\mid\psi(x)\leq y\leq\phi (x)\} \] Moreover, \(\psi (x)=\phi (x)\) on a dense set of values of \(x\). – Some consequences of this result are given and applied to the map \[ T(x,y)=(x+\omega , B\cos 2\pi x\tanh y) \] [C. Grebogi, E. Ott, S. Pelikan and J. A. Yorke, Physica D 13, 261-268 (1984; Zbl 0558.58036)].