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A random system with complete connections associated with a generalized Gauss-Kuzmin operator. (English) Zbl 0865.11055
Denote by $$C(0,1)$$ the Banach space of complex continuous functions of a real variable $$t\in[0,1]$$ (with the sup-norm). For $$f\in C(0,1)$$ we put $(G_\alpha f)(w) = \sum^\infty_{x=1} {\alpha^2 \over (\alpha x+w) (\alpha x+ \alpha-1+w)} \cdot f\left({\alpha \over \alpha x+ \alpha-1+w} \right)$ $$(w\in [0,1])$$ [cf. W. Fluch, Anz. Österr. Akad. Wiss. Math.-Natur. Kl. 124 (1987), 73-76 (1988; Zbl 0709.11040)].
In the paper a random system with complete connections is associated with the operator $$G$$. Investigating its ergodic properties the author obtains some results of the paper of Fluch (loc. cit.).

##### MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28D10 One-parameter continuous families of measure-preserving transformations