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arXiv · 0912.2329

The entropy of alpha-continued fractions: numerical results

Carlo Carminati·Stefano Marmi·Alessandro Profeti·Giulio Tiozzo
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Abstract

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

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Carlo Carminati, Stefano Marmi, Alessandro Profeti, Giulio Tiozzo. 2009-12-11. The entropy of alpha-continued fractions: numerical results. https://doi.org/10.1088/0951-7715%2F23%2F10%2F005

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