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Maximal \(S\)-expansions are Bernoulli shifts. (English) Zbl 0795.11030

Let \([a_ 0; \varepsilon_ 1 a_ 1,\dots]\) be a semi-regular continued fraction (SRCF) expansion of \(x\), i.e. \(\varepsilon_ i+ a_ i\geq 1\), \(\varepsilon_{i+1}+ a_ i\geq 1\), for \(i\geq 1\), \(\varepsilon_{i+1}+ a_ i\geq 2\) infinitely often and \(x=x_ 0= a_ 0+ \varepsilon_ 1 x_ 1^{-1},\dots, x_ i=a_ i+ \varepsilon_ n x_{i+1}^{-1}\). Finite truncation yields the sequence of convergents \(r_ n/s_ n= [a_ 0; \varepsilon_ 1 a_ 1,\dots, \varepsilon_ n a_ n]\) which converges to \(x\). If \(a_{k+1}= 1\), \(\varepsilon_{k+1}= \varepsilon_{k+2}=1\), then the expansion of \(x\) can be replaced by \[ [a_ 0; \varepsilon_ 1 a_ 1,\dots, \varepsilon_{k-1} a_{k-1}, \varepsilon_ k(a_ k+1), -(a_{k+2}+1), \varepsilon_{k+3} a_{k+3},\dots], \] which is again a SRCF expansion of \(x\). This operation is called by the author the singularization of the partial quotient \(a_{k+1}=1\). The regular continued fraction (RCF) expansion corresponds to \(\varepsilon_ i=1\) for all \(i\geq 1\) and is related to the Gauss dynamical system, namely \((T,[0,1]^ 2,\mu)\) with \(T(x,y)=( \{1/x\}, (y+[1/x])^{-1})\) and \(\mu(dx)= (\log 2)^{-1} (1+xy)^{-2} dx\), where \(\{x\}\) denotes the fractional part of \(x\) and \([x]= x- \{x\}\).
Many classical continued fractions (e.g. the nearest integer continued fraction (NICF), the Minkowski’s diagonal continued fraction, the optimal one, the Nakada \(\alpha\)-expansions) are derived from the strategy for singularization of the RCF which leads the author to introduce the notion of singularization area \(S\) with respect to \(T\) and the corresponding \(S\)-expansion system given by the induced transformation of \(T\) on \([0,1]^ 2\setminus S\). For more details, see the author [Acta Math. 57, 1-39 (1991; Zbl 0721.11029)]. The \(S\)-expansion is said to be maximal if \(\mu(S)\) is maximal (namely, \(\mu(S)= 2- {{\log(\sqrt{5}+1)} \over {\log 2}})\). This is the case for the \(\Sigma\)-expansion given by \(\Sigma= [{1\over 2},\;)\times [0, {{\sqrt{5}-1} \over 2}]\) and which corresponds to the NICF expansion.
The main result of this paper says that any maximal \(S\)-expansion system \({\mathcal S}\) is isomorphic to a Bernoulli shift. The proof exhibits an isomorphism between \({\mathcal S}\) and the above \(\Sigma\)-expansion system which is known to be a Bernoulli shift from a result due to G. L. Rieger [J. Reine Angew. Math. 310, 171-181 (1979; Zbl 0409.10038)].

MSC:

11K50 Metric theory of continued fractions
28D05 Measure-preserving transformations
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References:

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