Symmetric standard factorization

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Table of contents

1 Some combinatorics on words 
1.1 Notation and preliminaries
1.2 Lyndon words
1.3 Standard Sturmian Words
1.4 Christoel words
1.4.1 Standard factorization
1.4.2 Christoel tree
1.5 Balance property
1.5.1 Balance matrix
1.5.2 Properties of the balance matrix
1.5.3 Christoel words with a and b not coprime
2 Algebraic view 
2.1 Algebraic denition
2.2 More about Christoel words
2.2.1 Continued fractions
2.2.2 Stern-Brocot tree
2.2.3 Second construction of the balance test matrix
3 Second order balanced matrix 
3.1 Second order balance matrix
3.2 Recursive construction of the second order balance matrix
3.2.1 Properties of the matrix Ua
3.2.2 General form of the second order balance matrix
3.3 The construction of Ua
3.3.1 The trivial cases : z 2 f0; 1g
3.3.2 The general case : z 2
4 Some results 
4.1 Second order balance of C( a b )
4.1.1 Renement of the value of 2( a b ) using the continued fraction
4.1.2 Fibonacci sequence
4.2 Abelian vectors
5 Symmetric standard factorization 
5.1 Symmetric standard factorization
5.1.1 Geometrical examples
6 Synchronization of Christoel words 
6.1 Synchronization of Christoel words
6.2 Vertical invariant
6.3 Seeds for two generators
6.4 Horizontal invariant
6.5 Case of three generators
6.5.1 Generators not relatively co prime
6.6 Relation between In;k and Ihgi
6.7 Equal generators
6.8 Distinct generators
6.8.1 Fraenkel’s seed
6.9 Relation between 2 and 3 generators
7 Convexity and digital convex polyominoes 
7.1 Introduction
7.1.1 Discrete geometry and digital space
7.2 Theoretical results
7.2.1 Perturbations on the WN paths
7.2.2 Denition of the split operator
7.2.3 Commutativity of the split operator