Ergodic theory of numbers / Karma Dajani, Cor Kraaikamp.

Format:

Book

Author/Creator:

Dajani, Karma.

Contributor:

Kraaikamp, Cor.

Series:

Carus mathematical monographs ; no. 29.

The Carus mathematical monographs ; no. 29

Language:

English

Subjects (All):

Ergodic theory.

Physical Description:

x, 190 pages : illustrations ; 22 cm.

Place of Publication:

Washington, DC : Mathematical Association of America, [2002]

Summary:

Introduction to ergodic theory of numbers for graduate students and researchers.

Contents:

1.1 Decimal expansions of rational numbers 2

1.2 Another look at the decimal expansion 5

1.3 Continued fractions 20

2 Variations on a theme (Other expansions) 33

2.1 n-ary expansions 33

2.2 Luroth series 36

2.3 Generalized Luroth series 41

2.4 [beta]-expansions 51

3 Ergodicity 57

3.1 The Ergodic Theorem 57

3.2 Examples of normal numbers 71

3.3 [beta]-transformations 73

3.4 Ergodic properties of the [beta]-expansion 76

3.5 Ergodic properties of continued fractions 80

4 Systems obtained from other systems 89

4.1 GLS-expansion and [beta]-expansion: A first glimpse at their connection 89

4.2 Induced and integral transformations 92

4.3 Natural extensions 98

4.4 Natural extension of the GLS transformation 101

4.5 Natural extension of the [beta]-transformation 104

5 Diophantine approximation and continued fractions 115

5.2 The natural extension of the regular continued fraction 125

5.3 Approximation coefficients revisited 126

5.4 Other continued fractions 134

5.5 A skew product related to continued fractions 151

6 Entropy 155

6.2 The Shannon-McMillan-Breiman Theorem and some consequences 169

6.3 Saleski's Theorem 175.

Notes:

Includes bibliographical references (pages 179-185) and index.

ISBN:

0883850346

OCLC:

50642468