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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.).

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