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El. knyga: Combinatorics, Automata and Number Theory

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Edited by (Université de Ličge, Belgium), Edited by (Université de Paris VII)
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This collaborative volume presents 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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This collaborative volume presents trends arising from the fruitful interaction between combinatorics on words, automata and number theory.
List of contributors
ix
Preface xi
Acknowledgements xix
1 Preliminaries
1(33)
V. Berthe
M. Rigo
1.1 Conventions
1(2)
1.2 Words
3(10)
1.3 Languages and machines
13(9)
1.4 Associated matrices
22(4)
1.5 A glimpse at numeration systems
26(1)
1.6 Symbolic dynamics
27(5)
1.7 Exercises
32(1)
1.8 Notes
33(1)
2 Number representation and finite automata
34(74)
Ch. Frougny
J. Sakarovitch
2.1 Introduction
34(3)
2.2 Representation in integer base
37(16)
2.3 Representation in real base
53(24)
2.4 Canonical numeration systems
77(7)
2.5 Representation in rational base
84(11)
2.6 A primer on finite automata and transducers
95(8)
2.7 Notes
103(5)
3 Abstract numeration systems
108(55)
P. Lecomte
M. Rigo
3.1 Motivations
108(9)
3.2 Computing numerical values and S-representations
117(5)
3.3 S-recognisable sets
122(16)
3.4 Automatic sequences
138(14)
3.5 Representing real numbers
152(6)
3.6 Exercises and open problems
158(2)
3.7 Notes
160(3)
4 Factor complexity
163(85)
J. Cassaigne
F. Nicolas
4.1 Introduction
163(1)
4.2 Definitions, basic properties, and first examples
163(3)
4.3 The theorem of Morse and Hedlund
166(2)
4.4 High complexity
168(3)
4.5 Tools for low complexity
171(10)
4.6 Morphisms and complexity
181(4)
4.7 The theorem of Pansiot
185(29)
4.8 Complexity of automatic words
214(1)
4.9 Control of bispecial factors
215(6)
4.10 Examples of complexity computations for morphic words
221(5)
4.11 Complexity computation for an s-adic family of words
226(13)
4.12 Exercises and open problems
239(9)
5 Substitutions, Rauzy fractals and tilings
248(76)
V. Berthe
A. Siegel
J. Thuswaldner
5.1 Introduction
248(2)
5.2 Basic definitions
250(9)
5.3 Tilings
259(14)
5.4 Ancestor graphs and tiling conditions
273(11)
5.5 Boundary and contact graphs
284(6)
5.6 Geometric coincidences
290(6)
5.7 Overlap coincidences
296(13)
5.8 Balanced pair algorithm
309(6)
5.9 Conclusion
315(1)
5.10 Exercises
316(1)
5.11 Notes
317(7)
6 Combinatorics on Bratteli diagrams and dynamical systems
324(49)
F. Durand
6.1 Introduction
324(1)
6.2 Cantor dynamical systems
325(1)
6.3 Bratteli diagrams
325(5)
6.4 The Bratteli-Vershik model theorem
330(6)
6.5 Examples of BV-models
336(15)
6.6 Characterisation of Strong Orbit Equivalence
351(6)
6.7 Entropy
357(3)
6.8 Invariant measures and Bratteli diagrams
360(4)
6.9 Eigenvalues of stationary BV-models
364(6)
6.10 Exercises
370(3)
7 Infinite words with uniform frequencies, and invariant measures
373(37)
S. Ferenczi
T. Monteil
7.1 Basic notions
374(2)
7.2 Invariant measures and unique ergodicity
376(7)
7.3 Combinatorial criteria
383(5)
7.4 Examples
388(7)
7.5 Counter-examples
395(10)
7.6 Further afield
405(1)
7.7 Exercises
406(3)
7.8 Note: Dictionary between word combinatorics and symbolic dynamics
409(1)
8 Transcendence and Diophantine approximation
410(42)
B. Adamczewski
Y. Bugeaud
8.1 The expansion of algebraic numbers in an integer base
412(15)
8.2 Basics from continued fractions
427(3)
8.3 Transcendental continued fractions
430(6)
8.4 Simultaneous rational approximations
436(7)
8.5 Explicit examples for the Littlewood conjecture
443(5)
8.6 Exercises and open problems
448(1)
8.7 Notes
449(3)
9 Analysis of digital functions and applications
452(53)
M. Drmota
P. J. Grabner
9.1 Introduction: digital functions
452(4)
9.2 Asymptotic analysis of digital functions
456(24)
9.3 Statistics on digital functions
480(19)
9.4 Further results
499(6)
10 The equality problem for purely substitutive words
505(25)
J. Honkala
10.1 Purely substitutive words and DOL systems
505(2)
10.2 Substitutive words and HDOL sequences
507(2)
10.3 Elementary morphisms
509(3)
10.4 Nearly primitive DOL systems
512(3)
10.5 Periodic and nearly periodic words
515(4)
10.6 A balance property for ω-equivalent 1-systems
519(4)
10.7 The equality problem for purely substitutive words
523(2)
10.8 Automatic words
525(1)
10.9 Complexity questions
526(1)
10.10 Exercises
527(3)
11 Long products of matrices
530(33)
V. D. Blondel
R. M. Jungers
11.1 The joint spectral characteristics
531(10)
11.2 Applications
541(9)
11.3 The finiteness property
550(2)
11.4 Approximation algorithms
552(6)
11.5 Conclusions
558(1)
11.6 Exercises
559(2)
11.7 Notes
561(2)
References 563(31)
Notation index 594(5)
General index 599
Valérie Berthé is 'Directeur de Recherche CNRS' in the Montpellier Laboratory of Informatics, Robotics, and Micro-electronics (LIRMM) at the University of Montpellier 2, France. Michel Rigo is a Professor in the Department of Mathematics at the University of Ličge, Belgium.
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