Natural extensions and entropy ofα-continued fractions

Abstract

We construct a natural extension for each of Nakada's α-continued fraction transformations and show the continuity as a function of α of both the entropy and the measure of the natural extension domain with respect to the density function (1 + xy) −2 . For 0 < α ≤ 1, we show that the product of the entropy with the measure of the domain equals π 2 /6. We show that the interval (3 − √ 5)/2 ≤ α ≤ (1 + √ 5)/2 is a maximal interval upon which the entropy is constant. As a key step for all this, we give the explicit relationship between the α-expansion of α − 1 and of α.

Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy

“…The matching condition (28) implies the following identification between of the orbits of the two endpoints α and α − 1. Note that (27) implies that the first m 0 steps of the (symbolic) itinerary of α is constant for all α in the same qumterval, and the same is true for the first m 1 steps of the orbit of m 1 . A condition of this kind is called strong cycle condition in [23]; see Section 6 for a more detailed comparison.…”

“…Another feature which is still unproved is the smoothness of entropy on matching intervals. This is due to the fact that the natural extension has no finite rectangular structure when α ranges in a matching interval (see [27]). We conjecture that, as in the case we examined in this paper, on a matching interval densities are piecewise continuous, with discontinuity points located on the forward images of the endpoints (before matching occurs), while the branches of these densities are fractional transformation which move smoothly with the parameter (see also [9], Conj.…”

On some symmetric multidimensional continued fraction algorithms

“…However, this domain is interesting only if it has strictly positive measure, which is not clear in a number of cases; even in that case, it can be very difficult to describe, as shown by the case of the japanese continued fraction, which has been very precisely studied, see for example [KSS12]. Remark also that a very well studied case, that of Rauzy induction, show the complications which can appear : if one uses exactly the procedure we describe in the present paper, one does not get a natural extension, because one gets a subset of lower dimension (given by the subspace called H(π) by Veech, see [Vee82]), and it is necessary to add other coordinates to get the natural extension.…”

On the bifurcation set of unique expansions

“…assigning to each α the measure theoretic entropy h µα (T α ), has countably many intervals on which it is monotonic. The complement of the union of these intervals in [0, 1], i.e., the bifurcation set of ψ denoted by F , has Lebesgue measure 0 (see [29] and [10]) and Hausdorff dimension 1 (see [9]). Moreover, in [9] the authors identified a homeomorphism between the set F and the set Λ \ {0} from (1.4), giving also a relation to the set U (1).…”