Cor Kraaikamp

From greedy to lazy expansions and their driving dynamics

Abstract In this paper we study the ergodic properties of non-greedy series expansions to non-integer bases β > 1. It is shown that the so-called ‘lazy’ expansion is isomorphic to the ‘greedy’ expansion. Furthermore, a class of expansions to base β > 1, β ∉ ℤ, ‘in between’ the lazy and the greedy expansions are introduced and studies. It is shown that these expansions are isomorphic to expansions of the form Tx = βx + α (mod 1).

Good approximations and continued fractions

Let (qn)n be the sequence of best approximation denominators of an irrational number a . The set of real numbers x for which qnx —► 0 (mod 1) is studied. It is shown that a number x belongs to ceZ (mod 1) if and only if a simple condition on the speed of the convergence related to an arithmetic property of a is satisfied. This set is uncountable whenever a has unbounded partial quotients.

Natural extensions and entropy of α-continued fractions

We construct a natural extension for each of Nakadas 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 π2/6. We show that the interval 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.

The natural extension of the β-transformation

For each real number β>1 the β-transformation is dened by Tβx = βx(mod1). In this paper the natural extension Tβ of the ergodic system underlying Tβ is explicitly given. Furthermore, it is shown that a certain induced system of this natural extension is Bernoulli. Since Tβ is weakly mixing, due to W. Parry, it follows from a deep result of A. Saleski that the natural extension is also Bernoulli, a result previously obtained by M. Smorodinsky.

Metrical theory for optimal continued fractions

Received November 12, 1987; revised March 13, 1989 The ergodic system underlying the optimal continued fraction algorithm is introduced and studied. In particular the distribution of the sequence B”(x)” z r, which measures how well a number x is approximated by its convergent% is derived for almost all irrational numbers.

Ergodic properties of a dynamical system arising from percolation theory

We consider the following dynamical system: take a d -dimensional real vectorwith positive coordinates. Now keep the smallest coordinate and subtract this one from the others, and iterate this process. When the starting vector is x we denote by x n the result after n iterations. It is shown that for almost all x , lim n →∞ x n ≠ 0 (the null vector). This is shown to be equivalent to the conjectured finiteness of an algorithm which produces the critical probability in a certain dependent percolation model.

The distribution of some sequences connected with the nearest integer continued fraction

Let An/Bn, n = 1,2,… denote the sequence of convergents of the nearest integer continued fraction expansion of the irrational number x, and defineΘn(x): Bn|Bnx − An|, n = 1,2,…. In this paper the distribution of the two-dimensional sequence (Θn(x), Θn+1(x)), n = 1,2,… is determined for almost all x. Various corollaries are obtained, for instance Sendovs analogue of Vahlens theorem for the nearest integer continued fraction. The present method is an extension of the work by H. Jager on the corresponding problem for the regular continued fraction expansion.

Multi-Scale and Multi-Resolution Stochastic Modeling of Subsurface Heterogeneity by Tree-Indexed Markov Chains

A new methodology is proposed to handle multi-scale heterogeneous structures. It can be of importance in the field of hydrogeology and for petroleum engineers who are interested in characterizing subsurface heterogeneity at various scales. The framework of this methodology is based on a coarse to fine scale representation of the heterogeneous structures on trees. Different depths in the tree correspond to different spatial scales in representing the heterogeneous structures on trees. On these trees a Markov chain is used to describe scale to scale transitions and to account for the uncertainty in the stochastically generated images.We focus in this work on the description and application of the methodology to synthetic data that are geologically realistic. The methodology is flexible. Conditioning on field data and measurements is straightforward. Non-stationary and stationary fields, compound and nested structures can be addressed.

Random \beta-expansions

In this paper, random expansions to non-integer bases \beta >1 are studied. For \beta s satisfying \beta^2=n\beta +k (with 1\leq k\leq n ) and \beta^n=\beta^{n-1}+\dotsb + \beta +1 the ergodic properties of such expansions are described.

Generalization of a theorem of Kusmin

A Gauss-Kusmin theorem for the natural extension of the regular continued fraction expansion is given. A generalization of a theorem by D. E. Knuth is obtained by similar techniques.