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

K. Dajani; C. Kalle

doi:10.1007/s10474-009-8212-0

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

K. Dajani^*, C. Kalle

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a0; a1; a2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.

Original language	English
Pages (from-to)	21-45
Number of pages	25
Journal	Acta Mathematica Hungarica
Volume	125
Issue number	1
DOIs	https://doi.org/10.1007/s10474-009-8212-0
Publication status	Published - 6 Jul 2009

Keywords

Absolutely continuous invariant measure
Greedy expansion
Natural extension

Access to Document

10.1007/s10474-009-8212-0

Cite this

@article{69ac8c7931234228b961e6f9265fb150,

title = "A natural extension for the greedy β-transformation with three arbitrary digits",

abstract = "We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a0; a1; a2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.",

keywords = "Absolutely continuous invariant measure, Greedy expansion, Natural extension",

author = "K. Dajani and C. Kalle",

year = "2009",

month = jul,

day = "6",

doi = "10.1007/s10474-009-8212-0",

language = "English",

volume = "125",

pages = "21--45",

journal = "Acta Mathematica Hungarica",

issn = "0236-5294",

publisher = "Springer",

number = "1",

}

TY - JOUR

T1 - A natural extension for the greedy β-transformation with three arbitrary digits

AU - Dajani, K.

AU - Kalle, C.

PY - 2009/7/6

Y1 - 2009/7/6

N2 - We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a0; a1; a2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.

AB - We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a0; a1; a2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.

KW - Absolutely continuous invariant measure

KW - Greedy expansion

KW - Natural extension

UR - http://www.scopus.com/inward/record.url?scp=70450253330&partnerID=8YFLogxK

U2 - 10.1007/s10474-009-8212-0

DO - 10.1007/s10474-009-8212-0

M3 - Article

SN - 0236-5294

VL - 125

SP - 21

EP - 45

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

IS - 1

ER -

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

Abstract

Keywords

Access to Document

Fingerprint

Cite this