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From Christoffel words to Markoff numbers

Author: Christophe Reutenauer
Publisher: Oxford : Oxford University Press, 2019.
Edition/Format:   Print book : English : First editionView all editions and formats
Summary:

This book looks to expand on the relationship between Christoffel words and Markoff theory. Part 1 focuses on the classical theory of Markoff, while part II explores the more advanced and recent  Read more...

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Additional Physical Format: Electronic version:
Reutenauer, Christophe.
From Christoffel words to Markoff numbers.
Oxford : Oxford University Press, 2019
(OCoLC)1038493963
Document Type: Book
All Authors / Contributors: Christophe Reutenauer
ISBN: 0198827547 9780198827542
OCLC Number: 1037023637
Description: xi, 156 pages : illustrations ; 24 cm
Contents: The Theory of Markoff1: Basics2: Words2.1: Tiling the plane with a parallelogram2.2: Christoffel words2.3: Palindromes2.4: Standard factorization2.5: The tree of Christoffel pairs2.6: Sturmian morphisms3: Markoff numbers3.1: Markoff triples and numbers3.2: The tree of Markoff triples3.3: The Markoff injectivity conjecture4: The Markoff property4.1: Markoff property for infinite words4.2: Markoff property for bi-infinite words5: Continued fractions5.1: Finite continued fractions5.2: Infinite continued fractions5.3: Periodic expansions yield quadratic numbers5.4: Approximations of real numbers5.5: Lagrange number of a real number5.6: Ordering continued fractions6: Words and quadratic numbers6.1: Continued fractions associated to Christoffel words6.2: Marko supremum of a bi-innite sequence6.3: Lagrange number of a sequence7: Lagrange numbers less than 37.1: From L(s) < 3 to the Marko property7.2: Bi-infinite sequences8: Markoff's theorem for approximations8.1: Main lemma8.2: Markoff's theorem for approximations8.3: Good and bad approximations9: Markoff's theorem for quadratic forms9.1: Indefinite real binary quadratic forms9.2: Infimum9.3: Markoff's theorem for quadratic forms10: Numerology10.1: Thirteen Markoff numbers10.2: The golden ratio and other numbers10.3: The matrices U(w) and Frobenius congruences10.4: Markoff quadratic forms11: Historical notesThe Theory of Christoel Words12: Palindromes and periods12.1: Palindromes12.2: Periods13: Lyndon words and Christoffel words13.1: Slopes13.2: Lyndon words13.3: Maximal Lyndon words13.4: Unbordered Sturmian words13.5: Equilibrated Lyndon words14: Stern-Brocot tree14.1: The tree of Christoffel words14.2: Stern-Brocot tree and continued fractions14.3: The Raney tree and dual words14.4: Convex hull15: Conjugates and factors15.1: Cayley graph15.2: Conjugates15.3: Factors15.4: Palindromes again15.5: Finite Sturmian words16: Free group on two generators16.1: Bases and automorphisms16.2: Inner automorphisms16.3: Christoffel bases16.4: Nielsen's criterion16.5: An algorithm for the bases16.6: Sturmian morphisms again17: Complements17.1: Other results on Christoffel words17.2: Lyndon words and Lie theory17.3: Music
Responsibility: Christophe Reutenauer.

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This book is a well-written introduction to a fascinating subtopic in mathematics that should be accessible to a graduate student or a research mathematician. Most subsections are a few pages in Read more...

 
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