## Ergodic Theory of Numbers, Volume 29This book is an introduction to the ergodic theory behind common number expansions, for instance decimal expansions, continued fractions and many others. The questions studied are dynamical as well as number theoretic in nature, and the answers are obtained with the help of ergodic theory. What it means to be ergodic and the basic ideas behind ergodic theory are explained along the way. The book is aimed at introducing students with sufficient background knowledge in real analysis to a 'dynamical way of thinking'. The subjects covered vary from the classical to recent research which should increase the appeal of this book to researchers working in the field. |

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### Contents

Variations on a theme Other expansions | 33 |

Ergodicity | 57 |

Systems obtained from other systems | 89 |

Diophantine approximation and continued fractions | 115 |

Entropy | 155 |

179 | |

187 | |

### Common terms and phrases

5-expansion algebra Bernoulli shift Bernoullicity Borel measure Borel set called consider continued fraction expansion continued fraction map convergents Corollary countable cylinder sets decimal expansion define definition denote entropy equal ergodic theory Euclid's algorithm eventually-periodic example Exercise exists full interval function Gauss measure given GLS expansion GLS-transformation GLS(I induced transformation infinite integer intervals of rank invariant measure isomorphic Jager Lebesgue measure Let x e Liiroth series limsup Lochs mathematics measure preserving transformation measure space measure zero n-ary expansions natural extension NICF normal numbers notation Notice obtained partial quotients partition element Pisot positive integer positive measure preserving with respect probability measure probability space proof pseudo-golden mean rational number real number regular continued fraction Remark result satisfies Section semi-algebra sequence of digits sets of measure Show singularization area SRCF-expansion subset symbols T-invariant Theorem