Substitutions in dynamics, arithmetics, and combinatorics /
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| Author / Creator: | Pytheas Fogg, N. |
|---|---|
| Imprint: | Berlin ; New York : Springer, 2002. |
| Description: | xvii, 402 p. : ill. ; 24 cm. |
| Language: | English |
| Series: | Lecture notes in mathematics, 0075-8434 ; 1794 Lecture notes in mathematics (Springer-Verlag) ; 1794. |
| Subject: | |
| Format: | E-Resource Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4745036 |
Table of Contents:
- Preface
- List of contributors
- 1. Basic notions on substitutions
- 1.1. Word combinatorics
- 1.2. Substitutions
- 1.3. Automata
- 1.4. Topological and measure-theoretic dynamical systems
- 1.5. Spectral theory
- 1.6. Factors of substitutive dynamical systems
- Part I. Arithmetics and combinatorics of substitutions
- 2. Substitutions, arithmetic and finite automata: an introducion
- 2.1. The Morse sequence
- 2.2. The Rudin-Shapiro sequence
- 2.3. The Baum-Sweet sequence
- 2.4. The Cantor sequence
- 2.5. An application of substitutions to criteria of divisibility
- 2.6. The Fibonacci sequence
- 3. Automatic sequences and transcendence (by V. Berthe)
- 3.1. Introduction
- 3.2. The Christol, Kamae, Mendès France, and Rauzy theorem
- 3.3. Transcendence in the real case and continued fractions
- 3.4. Some functions defined by Carlitz
- 3.5. Some examples of automaton proofs
- 3.6. Conclusion
- 4. Substitutions and partitions of the set of positive integers
- 4.1. Beatty and Sturmian sequences, and associated partitions
- 4.2. Partitions given by substitutions
- 4.3. Similis partitions
- 4.4. Log-fixed points and Kolakoski words
- 4.5. Problems
- Part II. Dynamics of substitutions
- 5. Substitutions and symbolic dynamical systems
- 5.1. The Morse sequence: elementary properties
- 5.2. The Morse sequence: advanced properties
- 5.3. The Rudin-Shapiro sequence
- 5.4. The Fibonacci sequence
- 5.5. The Chacon sequence
- 6. Sturmian Sequences
- 6.1. Sturmian sequences. Basic properties
- 6.2. Biinfinite Sturmian sequences
- 6.3. Coding sequences for Sturmian sequences and Sturmian systems160
- 6.4. Sturmian sequences: arithmetic properties; continued fractions 175
- 6.5. Sturmian substitutions. Dynamical interpretations
- 6.6. Natural extension for the recoding of Sturmian sequences
- 6.7. Miscellaneous remarks
- 7. Spectral theory and geometric representation of substitutions
- 7.1. Shifts of finite type: introduction
- 7.2. Substitutive dynamical systems and shifts of finite type
- 7.3. Spectral theory of substitutive dynamical systems
- 7.4. The Rauzy fractal
- 7.5. Geometric realization of substitutions of Pisot type
- 7.6. Extensions and applications
- 8. Diophantine approximations, substitutions, and fractals
- 8.1. Substitutions and domains with fractal boundary
- 8.2. Generalized substitutions
- 8.3. Dynamical systems associated with the stepped surface
- 8.4. Renormalization and realization of substitutive systems
- 8.5. Fractal boundary
- 8.6. Continued fraction expansions and substitutions
- 8.7. Diophantine applications
- Part III. Extensions to free groups and interval transformations
- 9. Infinite words generated by invertible substitutions
- 9.1. Preliminary
- 9.2. Structure of invertible substitutions
- 9.3. Singular words of the Fibonacci word and applications
- 9.4. Properties of factors of the fixed points of invertible substi- tutions
- 10. Polynomial dynamical systems associated with substitutions
- 10.1. Polynomial identities in the algebra of 2 × 2-matrices
- 10.2. Trace maps
- 10.3. The case of 3 × 3-matrices
- 10.4. Comments
- 11. Piecewise linear transformations of the unit interval and Cantor sets
- 11.1. Definitions
- 11.2. Ergodic properties of piecewise Markov transformations
- 11.3. Non-Markov transformation: ß-expansion
- 11.4. Cantor sets
- 12. Some open problems
- 12.1. The S-adic conjecture
- 12.2. Multidimensional continued fraction expansions
- 12.3. Combinatorics on two-dimensional words
- 12.4. Substitutions, rotations and toral automorphisms
- 12.5. Arithmetics in and
- A. Undecomposable matrices in dimension 3
- References
- Index
