## Combinatorics, Automata and Number TheoryThis collaborative volume presents recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Topics include numeration systems, word complexity function, morphic words, Rauzy tilings and substitutive dynamical systems, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic properties of digital functions, decidability issues for D0L systems, matrix products and joint spectral radius. Topics are presented in a way that links them to the three main themes, but also extends them to dynamical systems and ergodic theory, fractals, tilings and spectral properties of matrices. Graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, fractals, tilings and stringology will find much of interest in this book. |

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### Contents

Number representation and ﬁnite automata | 34 |

Abstract numeration systems | 108 |

Factor complexity | 163 |

Substitutions Rauzy fractals and tilings | 248 |

Combinatorics on Bratteli diagrams and dynamical systems | 324 |

Inﬁnite words with uniform frequencies | 373 |

Transcendence and Diophantine approximation | 410 |

Analysis of digital functions and applications | 452 |

The equality problem for purely substitutive words | 505 |

Long products of matrices | 530 |

563 | |

594 | |

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### Common terms and phrases

algebraic alphabet assume automaton balanced pair bispecial factors bounded Chapter clopen set coincidence condition combinatorial complexity function compute consider continued fraction converges Corollary deﬁned Deﬁnition denote diﬀerent digits dynamical system eigenvalue element endomorphism eventually periodic Example Exercise exists expansion Fibonacci Fibonacci word ﬁnite ﬁnite automata ﬁnite word ﬁrst given Hence implies infinite inﬁnite word invariant measures joint spectral radius label Lemma letter linear recurrence minimal morphic word morphism multiple tiling non-empty normalisation numeration system overlap coincidence Pisot irreducible substitution Pisot number polynomial positive integer preﬁx Proof Let Proposition prove purely morphic rational Rauzy real number regular language representation result S-recognisable satisﬁes satisfying Section sequence set of matrices spectral radius Sturmian words subset subshift substitutive words symbolic dynamical tiling property topological transducer uniform frequencies uniformly recurrent uniquely ergodic unit Pisot irreducible words of length