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18 results

Ergodic Theory of Numbers
  • Karma Dajani, Cor Kraaikamp
  • Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 2002
  • Ergodic Theory of Numbers is an introduction to the ergodic theory behind common number expansion, like decimal expansions, continued fractions, and many others. However, its aim does not stop there. For undergraduate students with sufficient background knowledge in real analysis and graduate students interested in the area, it is also an introduction to a "dynamical way of thinking." The questions studied here are dynamical as well as number theoretical in nature, and the answers are obtained with the help of ergodic theory. Attention is focused on concepts like measure-preserving, ergodicity, natural extension, induced transformations, and entropy. These concepts are then applied to familiar expansions to obtain old and new results in an elegant and straightforward manner. What it means to be ergodic and the basic ideas behind ergodic theory will be explained along the way. The subjects covered vary from classical to recent, which makes this book appealing to researchers as well as students.

  • Mixing properties of (α,β)-expansions

  • KARMA DAJANI, YUSUF HARTONO, COR KRAAIKAMP
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 29 / Issue 4 / August 2009
    Published online by Cambridge University Press:
    01 August 2009, pp. 1119-1140
    Print publication:
    August 2009

  • Let 0<α<1 and β>1. We show that every x∈[0,1] has an expansion of the form where hi=hi(x)∈{0,α/β}, and pi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values of α, we give the relationship of this expansion with the greedy β-expansion.

  • Preface

  • Karma Dajani, University of Utrecht, Cor Kraaikamp, Delft University of Technology
  • Book:
    Ergodic Theory of Numbers
    Published by:
    Mathematical Association of America
    Published online:
    05 June 2014
    Print publication:
    01 June 2002, pp ix-x

  • Summary

    In this book we will look at the interaction between two fields of mathematics: number theory and ergodic theory (as part of dynamical systems). The subject under study is thus part of what is known in France as Théorie Ergodique des Nombres, and consists of a family of series expansions of numbers in the unit interval [0, 1] with their ‘metrical properties.’ So the questions we want to study are number theoretical in nature, and the answers will be obtained with the help of ergodic theory. That is, we will view these expansions as iterations of an appropriate measure-preserving transformation on [0, 1], which will then be shown to be ergodic. The number-theoretical questions will be reformulated in the language of ergodic theory. What it means to be ergodic, or—in general—what the basic ideas behind ergodic theory entail, will be explained along the way.

    This book grew out of a course given in 1996 at George Washington University, Washington, DC, during the Summer Program for Women in Mathematics, sponsored by NSA. Our aim was not to write yet another book on ergodic theory (there are already several outstanding books, most of them mentioned in these pages), but to introduce first-year graduate students to a dynamical way of thinking. Consequently, many classical concepts from ergodic theory are either briefly mentioned, or even left out.

  • On normal numbers for continued fractions

  • COR KRAAIKAMP, HITOSHI NAKADA
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 20 / Issue 5 / October 2000
    Published online by Cambridge University Press:
    10 November 2000, pp. 1405-1421
    Print publication:
    October 2000

  • In this paper we will show that any number normal with respect to the regular continued fraction (RCF) is also normal with respect to the continued fraction from below (BCF) and the nearest integer continued fraction (NICF). Furthermore, we will show that in case of the NICF a number is normal with respect to the RCF-expansion if and only if it is NICF-normal. This can be generalized to a larger class of semi-regular continued fraction expansions.

  • Ergodic properties of a dynamical system arising from percolation theory

  • Cor Kraaikamp, Ronald Meester
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 15 / Issue 4 / August 1995
    Published online by Cambridge University Press:
    19 September 2008, pp. 653-661
    Print publication:
    August 1995

  • 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 xn the result after n iterations. It is shown that for almost all x, limn→∞xn ≠ 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.