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On the metrical theory of a peculiar continued fraction expansion. (English) Zbl 1174.11065

Let \(I = [0, 1]\) be the unit interval and let \(T\) be the transformation on \(I\) defined by \[ T(x) = 1_{\text{[0, 1/2)}}(x){x \over {1-x}} + 1_{\text{[1/2, 1]}}(x){{1-x}\over x}, \;\; x \in I \] If \(\mathcal B_{\text I}\) is the \(\sigma\)-algebra of Borel subsets of \(I\), consider the sequence \((\varepsilon_ n)_{n \in \mathbb N ^*}\), \(\mathbb N^* = \{1, 2, \dots \}\), of \(\{0, 1\}\)-valued random variables on \((I, \mathcal B_{\text I})\), where \(\varepsilon_n = \varepsilon_1 \circ T^{n-1}\) with \(\varepsilon_1 = 1_{\text{[1/2, 1]}}\) and \(T^0(x)= x\), \(x \in I\). Also, consider the random variables \(z_n\), \(n \in \mathbb N ^*\), defined recursively by \(z_n = 1 + (1-2\varepsilon_n)/ (\varepsilon_n + 1/z_{n-1})\) with arbitrary \(z_0\). The authors prove that for any \(n \in \mathbb N^*\), the conditional probability \(\lambda(\varepsilon_{n+1} | \varepsilon_1, \dots, \varepsilon_n)\) is equal to \((z_n + 1)/(z_n + 2)\) when \(z_0 = 0\). Also, for an arbitrary \(z_0\), \((z_n)_{n \in \mathbb N ^*}\) is a \([0, \infty)\)-valued Markov process with transition probability function \[ P(z, A) = {1 \over{z+2}} 1_{\text A} \left({1 \over{z+1}}\right) + {{z+1} \over{z+2}} 1_{\text A} (z+1), \;\; z \in [0, \infty), \; A \in \mathcal B_{[0, \infty)} \] that has as stationery distribution the \(\sigma\)-finite, infinite measure with density \(1/(z+1)\) on \([0, \infty)\).

MSC:

11K50 Metric theory of continued fractions
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
47B38 Linear operators on function spaces (general)
60J05 Discrete-time Markov processes on general state spaces
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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