An infinite-order-chain representation of the Hurwitz singular continued fraction expansion (Q2736608)

Let \(g < 1\), \(g ^{2} = 1+ g\) and \(\tau_{g}': [ g-1, g) \rightarrow [ g-1, g)\) be defined as \(\omega \rightarrow |\omega ^{- 1} |-[ |\omega ^{- 1} |+ 1-g ]\) (\(0\) for \(\omega = 0)\), let \(\Omega_{g} = [ g-1, g) \times [ 0, 1/ 2 ]\), \(\tau_{g} (\omega, \theta) =\) (\(\tau_{g}'(\omega), ([ |\omega ^{- 1} |+ g ^{2} ]+ \text{sgn} (\omega) \theta) ^{- 1})\), \(\varepsilon_{g, 1} (\omega, \theta) = \text{sgn} (\omega)\), \(a_{g, 1} (\omega, \theta) = [ |\omega ^{- 1} |+ 1-g ]\), \(\varepsilon_{g, n+ 1} = \varepsilon_{g, 1} \circ \tau_{g} ^{n}\), \(a_{g, n+ 1} = a_{g, 1} \circ \tau_{g} ^{n}\). One remarks that for \(n \leq 0\) these depend only on \(\theta\). Let \(\mu_{g}\) be the probability on \(\Omega_{g}\) with a density proportional to \((1+ x y) ^{- 1}\). The author proves that the conditional distribution of \((\omega, \theta) \rightarrow \omega\), on \((\Omega_{g}, \cdot, \mu_{g})\), with respect to \((\varepsilon_{g, n}, a_{g, n})_{n\leq 0}\), charges \([ g-1, x)\) with \((x+ g ^{2}) (a g+ 1)/ (a x+ 1)\), where \(a\) is the continued fraction \(\varepsilon_{g, 0}/ (a_{g, 0}+ \varepsilon_{g, 1}/ (a_{g, 1}+ \varepsilon_{g, 2}/ (\ldots)))\). A less obvious corollary is a formula for the conditional probability of a given value of \(\varepsilon_{g, 1}\), \(a_{g, 1}\). Using these results, the author constructs a \([ 0, 1/ 2 ]\)-valued Markov chain on \([ g-1, g)\), with transitions from an \(s\) only to \((i+ k s) ^{- 1}\), \(i > 1\) integer, \(k = \pm 1\), with explicitly expressed transition probability, and discusses the relations of these topics with chains with complete connections.