The entropy of alpha-continued fractions: numerical results
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.
Keywords
Cite
@article{arxiv.0912.2329,
title = {The entropy of alpha-continued fractions: numerical results},
author = {Carlo Carminati and Stefano Marmi and Alessandro Profeti and Giulio Tiozzo},
journal= {arXiv preprint arXiv:0912.2329},
year = {2015}
}
Comments
33 pages, 14 figures