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On the ergodic behaviour of a random system with complete connections associated with a concrete piecewise fractional linear map. (English) Zbl 1161.37013
The author is concerned with the well-known transformation of the unit interval [0,1] defined by \(S(x)=x/(1-x)\) for \(0\leq x\leq 1/2\) and \(S(x)=(1-x)/x\) for \(1/2\leq x\leq 1\) that is known to have a unique infinite, \(\sigma\)-finite, absolutely continuous invariant measure with density \(1/x,\;0<x\leq 1\).
She associates with this transformation a random system with complete connection that is claimed to be with contraction [cf. M. Iosifescu and S. Grigorescu, Dependence with complete connections and its applications. Cambridge University Press (1990; Zbl 0749.60067) (paperback edition 2009), especially pages 79–80]. This is not true since (see page 151 of the paper) \(r_{1}=1\), but not \(r_{1}<1\) as asserted.
It is also strange that the author does not realize that for \( 0\leq y\leq 1\) the sums of the series \(\sum_{j=0}^{\infty }\left[ \frac{1}{1+(j+1)y}-\frac{1}{1+(j+2)y}\right] ,\;\sum_{j=0}^{\infty }\left[ \frac{y}{y+j+1}-\frac{y}{y+j+2}\right] \) and \( \sum_{j=0}^{\infty }\left[ \frac{1}{1+jy}- \frac{1}{1+(j+1)y}\right] \) are \(\frac{1}{y+1},\;\frac{y}{y+1}\) and \(1\), respectively.

MSC:
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60K99 Special processes
60A10 Probabilistic measure theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G10 Stationary stochastic processes
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