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An entropy problem of the \(\alpha \)-continued fraction maps. (English) Zbl 1492.11116

The notion of \(\alpha\)-continued fraction map was introduced by the author in [Tokyo J. Math. 4, 399–426 (1981; Zbl 0479.10029)]. The main result of this paper states that the maximum value of the entropy of \(\alpha\)-continued fraction maps is \(\frac{1}{\log(g+1)}\frac{\pi^2}{6}\) and the entropy with respect to the absolutely continuous invariant probability measure is \(\frac{1}{\log(g+1)}\frac{\pi^2}{6}\) if and only if \(g^2\le\alpha\le g\) with \(g=(\sqrt{5}-1)/2\). The proof of this result is to apply the idea of the geodesic continued fractions introduced in [C. Kraaikamp et al., Nonlinearity 25, No. 8, 2207–2243 (2012; Zbl 1333.11079)], to the metric theory of the \(\alpha\)-continued fractions. With this idea, the main result gives the answer to the question in [A. F. Beardon et al., Mich. Math. J. 61, No. 1, 133–150 (2012; Zbl 1382.11023)] concerning the maximum value of the entropy of the \(\alpha\)-continued fraction maps.

MSC:

11J70 Continued fractions and generalizations
11K50 Metric theory of continued fractions
37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
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References:

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