The natural extension of the ?-transformation

Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

“…In [DKS96], Dajani et al provide an explicit construction of the natural extension of the β-transformation for any β > 1 in dimension three, the third dimension being given by the height in a stacking structure. This construction is minimal in the above sense.…”

A natural extension for the greedy β-transformation with three arbitrary digits

“…The transformation T β has a unique invariant measure, absolutely continuous with respect to λ. Rényi proved the existence of this measure in [14], and Gel'fond and Parry independently, gave an explicit formula for the density function of this measure in [7] and [11] respectively. The invariant measure has the unit interval [0, 1) as its support and the density function h c is given as (4) h c : [0, 1) → [0, 1) :…”

“…Throughout the rest of the paper, we will assume that 1 < β < 3, and A = {0, 1, u} is an allowable set. Our construction resembles the version built in [4] for the classical greedy β-transformation (see also [1]). The domain of the natural extension is roughly a (union) of rectangular regions in R 2 , with invariant measure the restriction of the 2-dimensional Lebesgue measure to our domain.…”

“…To illustrate the construction of our version of the natural extension, an example of a specic greedy β-transformation with three digits can be found in the last section. Here β is the golden mean and A = { 0, 1, 4 3 } .…”

Dynamical directions in numeration

“…Let us note that the natural extension of the transformation T (in the fibred case) is a useful tool to find explicitly invariant measures. It is standard for continued fractions; see for example [90] for more special continued fractions, and [111] for the β-transformation (see also Question 2.19 below). See also [78] (and the bibliography therein) for recent results on the camparison between the distribution of the number of digits determined when comparing two types of expansions in integer bases produced by fibred systems (e.g., continued fractions and decimal expansions).…”