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Ergodic properties of skew products with Lasota-Yorke type maps in the base. (English) Zbl 0815.28013

Summary: We consider skew products \(T(x, y)= (f(x), T_{e(x)}y)\) preserving a measure which is absolutely continuous with respect to the product measure. Here \(f\) is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and \(T_ i\), \(i= 1,\dots,\max e\), are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for \(T\) and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of \(T\). We apply these results to random perturbations.

MSC:

28D05 Measure-preserving transformations
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