Ergodic properties of generalized Lüroth series

The natural extension of the ?-transformation

“…In this paper we will obtain Smorodinsky's result by explicitly constructing the natural extension of ([0,1 ), B, v~, T~) (Section 2), and showing that a certain induced transformation of this natural extension is isomorphic to a Bernoulli automorphism, which is completely u~derstood (see also Section 3 or [1]). In Section 4 we then show that Smorodinsky's result follows from Parry's observation that TZ is weakly-mixing, and a theorem by A. Saleski [111.…”

“…An easy example. In case fl > 1 is the positive root of the polynomial X '~ -X m-1 -... -X -1, m >= 2, one easily sees I that the natural extension of ([0,1),B, vz,T~) is given by (Xz, B, See also [1]. For vrt = 2 one has thv, t fl---s2-~-)-, which is the {[olden mean.…”

“…In [1] the so-called Generalized Liiroth Series (shortly GLS) were introduced and their ergodic properties studied. We recall here some of the basic definitions and properties of GLS-expansions, and then we will show how GLS-expansions can be used to obtain a deeper understanding of the natural extension of the 3-transformation T/~.…”

“…This transformation gives rise to the/~-expansion introduced by A. R~nyi [9]: for any 0 _< x _< 1, (1) x = ~ dk/3 -k, When/3 is an integer, (1) is the usual digit expansion with base/3. In this case T/~ is clearly related to the Bernoulli-shift on/~ symbols; however, for /3 ~ Z the situation is more complicated.…”

The efficiency of approximating real numbers by Lüroth expansion

“…The behavior of approximating real numbers by the Lüroth expansion was thoroughly investigated in [2], [3], where the authors studied the distribution of the approximation coefficients θ n = θ n (x) = Q n (x)x − P n (x) for n 1. The errorsum function of the Lüroth expansion defined by S(x) = ∞ n=1 (x − P n (x)/Q n (x)) was studied in [13], where the authors investigated the properties of this function and determined the Hausdorff dimension of its graph.…”