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Homoclinic classes and finitude of attractors for vector fields on $$n$$-manifolds. (English) Zbl 1035.37007
The author defines the notion of the almost sure rotation number for certain expanding piecewise affine endomorphisms of the circle as the rotation number for almost every point of the circle. It is shown, that the dependance of the almost sure rotation number on the parameters is of Hölder type with any exponent $$\alpha\in (0,1)$$ but not Lipschitzian. The set of parameters giving an irrational almost sure rotation number is of Hausdorff dimension one. The method of the symbolic Perron-Frobenius operator is applied.

##### MSC:
 37B25 Stability of topological dynamical systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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