The entropy of α-continued fractions: numerical results
Abstract
We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as α-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 \mathcal{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 \mathcal{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 \mathcal{M} on which it is not even locally monotone.
- Publication:
-
Nonlinearity
- Pub Date:
- October 2010
- DOI:
- 10.1088/0951-7715/23/10/005
- arXiv:
- arXiv:0912.2329
- Bibcode:
- 2010Nonli..23.2429C
- Keywords:
-
- Mathematics - Dynamical Systems;
- Mathematics - Number Theory;
- 11K50;
- 37A10 (Primary);
- 37A35;
- 37E05 (Secondary)
- E-Print:
- 33 pages, 14 figures