Orbits of N-expansions with a finite set of digits

For N∈ N_{≥ 2} and α∈ R such that 0<α≤N-1, we define I_α: = [α, α+ 1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→Iα-, which is defined as Tα(x):=Nx-d(x), where d: I_α→ N is defined by d(x):=⌊Nx-α⌋. For N∈ N_{≥ 7}, for certain values of α, open intervals (a, b) ⊂ I_α exist...

Natural extensions for Nakada's α-expansions: Descending from 1 to g²

By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension...

Invariant measures for continued fraction algorithms with finitely many digits

In this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss–Kuzmin...

Pierre Liardet 1943–2014

Understanding the Non-Gaussian Nature of Linear Reactive Solute Transport in 1D and 2D: From Particle Dynamics to the Partial Differential Equations

In the present study, we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this...

Mixing properties of (alpha, beta)-expansions

Metric Properties of Denjoy’s Canonical Continued Fraction Expansion

Metric properties of Denjoy's canonical continued fraction expansion are studied, and the natural extension of the underlying ergodic system is given. This natural extension is used to give simple proofs of results on mediant convergents obtained by W. Bosma in 1990.

Asymptotics: Particles, Processes and Inverse Problems. Festschrift for Piet Groeneboom

Entropy quotients and correct digits in number-theoretic expansions

Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the...

On Denjoy's canonical continued fraction expansion

A Note on Hurwitzian Numbers

In this note Hurwitzian numbers are defined for the nearest integer, and backward continued fraction expansions, and Nakada's $\alpha$-expansions. It is shown that the set of Hurwitzian numbers for these continued fractions coincides with the classical set of such numbers.

Metric Properties of Denjoy's Canonical Continued Fraction Expansion

Metric properties of Denjoy's canonical continued fraction expansion are studied, and the natural extension of the underlying ergodic system is given. This natural extension is used to give simple proofs of results on mediant convergents obtained by W. Bosma in 1990.