Combinatorial properties of f -palindromes Sébastien Labbé Aims of the Combinatorial properties of f -palindromes talk Definitions and notations Hof, Knill and Sébastien Labbé Simon Conjecture Main Results LaCIM, Université du Québec à Montréal Further work 25 mai 2009
Outline Combinatorial properties of f -palindromes Sébastien Aims of the talk 1 Labbé Aims of the talk Definitions and notations 2 Definitions and notations Hof, Knill and Hof, Knill and Simon Conjecture 3 Simon Conjecture Main Results Main Results Further work 4 Further work 5
Aims of the talk Combinatorial properties of f -palindromes Sébastien Labbé Hof, Knill and Simon conjectured in 1995 a Aims of the characterization of the fixed point of morphisms having talk an infinite palindrome complexity (the number of Definitions and notations palindrome factors). Hof, Knill and Simon Recently, this conjecture was solved for the binary Conjecture alphabet (Tan, 2007). Main Results Further work We show a similar result for fixed points of uniform morphisms having an infinite number of f -palindromes.
Definitions and notations Combinatorial properties of f -palindromes A set Σ called alphabet whose elements are called Sébastien Labbé letters. Elements w of the free monoid Σ ∗ are called words. We Aims of the talk note w ∈ Σ ∗ and Definitions and notations w = w 0 w 1 w 2 · · · w n − 1 , w i ∈ Σ . Hof, Knill and Simon Conjecture The length of w is | w | = n . Main Results An infinite word w = w 0 w 1 · · · is a map w : N → Σ . Further work If w = pfs , then p is called a prefix, f a factor and s a suffix of w . Fact ( w ) is the set of the (finite) factors of w .
Definitions and notations Combinatorial properties of f -palindromes Sébastien Labbé Aims of the The reversal of a finite word w talk Definitions � w = w n − 1 w n − 2 · · · w 1 w 0 . and notations Hof, Knill and Simon A palindrome is a word w such that w = � w . Conjecture Main Results Pal ( w ) = Fact ( w ) ∩ Pal (Σ ∗ ) is the set of the palindrome Further work factors of w .
Definitions and notations Combinatorial A morphism is a function ϕ : Σ ∗ → Σ ∗ such that properties of f -palindromes Sébastien u , v ∈ Σ ∗ . ϕ ( uv ) = ϕ ( u ) ϕ ( v ) for all Labbé Aims of the A morphism ϕ is primitive if there exists k ∈ N such that talk every letters of Σ appear in ϕ k ( α ) for all α ∈ Σ . Definitions and notations A morphism is uniform if | ϕ ( α ) | = | ϕ ( β ) | for all α, β ∈ Σ . Hof, Knill and Simon ϕ the morphism defined by α �→ � Conjecture We denote by � ϕ ( α ) . Main Results A fixed point of a morphism ϕ is a word w such that Further work ϕ ( w ) = w . We say that ϕ is a right-conjugate of ϕ ′ if there exists a ord u ∈ Σ ∗ such that ϕ ( α ) u = u ϕ ′ ( α ) , for all α ∈ Σ .
Definitions and notations Combinatorial Example properties of f -palindromes The non primitive morphism defined on Σ = { a , b , c , d , e } Sébastien Labbé by a �→ ab , b �→ ba , c �→ cd , d �→ c , e �→ e has two finite fixed points : Aims of the talk ε , the empty word Definitions and notations e Hof, Knill and Simon and three infinite fixed points : Conjecture abbabaabbaababbaabba · · · Main Results Further work baababbaabbabaabbaab · · · cdccdcdccdccdcdccdcdccdccdcdccdccdcdccdc · · · Each fixed point may be obtained by considering n →∞ ϕ n ( α ) , α ∈ Σ . lim
About palindrome complexity Combinatorial properties of Proposition (Droubay, Justin, Pirillo, 2001) f -palindromes Let w be a finite word. Then, Sébastien Labbé | Pal ( w ) | ≤ | w | + 1 Aims of the talk Definitions and Sturmian words reach that bound. and notations Hof, Knill and Simon Definition (Brlek, Hamel, Nivat, Reutenauer, 2004) Conjecture Main Results Let w be a finite word. The defect D ( w ) of w is Further work D ( w ) = | w | + 1 − | Pal ( w ) | . and w is full if D ( w ) = 0. Moreover, the defect of a infinite word is the supremum of the defect of its finite prefixes. Full words are also called rich in the recent litterature.
Hof, Knill and Simon Conjecture Combinatorial properties of f -palindromes Sébastien Labbé Aims of the talk Definition (Hof, Knill and Simon, 1995) Definitions and notations A morphism ϕ is in class P if there exists a palindrome p Hof, Knill and and for each α ∈ Σ there exists a palindrome q α such that Simon Conjecture ϕ ( α ) = pq α . Main Results Further work
Hof, Knill and Simon Conjecture Combinatorial properties of f -palindromes The morphism Sébastien Labbé ϕ : { a , b } ∗ → { a , b } ∗ Aims of the talk �→ bb · aba a Definitions b �→ bb · a and notations Hof, Knill and is in class P . It has only one fixed point beginning by letter b . Simon Conjecture Main Results Further work i 0 1 2 3 4 5 6 7 | ϕ i ( a ) | 1 5 19 71 265 989 3691 13775 | Pal ( ϕ i ( a )) | 2 6 20 72 266 990 3692 13776
Hof, Knill and Simon Conjecture Combinatorial properties of f -palindromes The square of the Thue-Morse morphism Sébastien Labbé µ : a �→ ab , b �→ ba is in class P : Aims of the µ 2 : { a , b } ∗ → { a , b } ∗ talk Definitions a �→ abba and notations b �→ baab Hof, Knill and Simon Conjecture The palindrome complexity table of one of its fixed point is : Main Results Further work i 0 1 2 3 4 5 6 7 | µ i ( a ) | 1 2 4 8 16 32 64 128 | Pal ( µ i ( a )) | 2 3 5 9 15 29 53 109
Hof, Knill and Simon Conjecture Combinatorial properties of f -palindromes The morphism Sébastien Labbé ϕ : { a , b } ∗ → { a , b } ∗ Aims of the �→ talk a abb Definitions b �→ ba and notations Hof, Knill and is not in class P . It has two infinite fixed points having both Simon Conjecture 23 palindromes : Main Results Further work i 0 1 2 3 4 5 6 7 8 | Pal ( ϕ i ( a )) | 2 4 8 15 23 23 23 23 23 | Pal ( ϕ i ( b )) | 2 3 6 13 18 23 23 23 23
Hof, Knill and Simon Conjecture Combinatorial properties of f -palindromes In their article, Hof, Knill and Simon also said : Sébastien “Clearly, we could include into class P Labbé substitutions of the form s ( α ) = q α p. We do not Aims of the talk know whether all palindromic x s arise from Definitions substitutions that are in this extended class P .” and notations Hof, Knill and Their quote is now called HKS Conjecture and it may be Simon Conjecture stated in the folowing way : Main Results Conjecture (Hof, Knill, Simon, 1995) Further work Let w be a fixed point of a primitive morphism. Then, | Pal ( w ) | = ∞ if and only if there exists a morphism ϕ such that ϕ ( w ) = w and such that either ϕ or � ϕ is in class P .
Hof, Knill and Simon Conjecture Combinatorial properties of f -palindromes Proposition (Blondin-Massé, 2007) Sébastien The morphism ϕ defined by a �→ abbab , b �→ abb is such Labbé ϕ are in class P but lim n →∞ ϕ n ( a ) has an that neither ϕ nor � Aims of the talk infinite number of palindromes. Definitions and notations Hence, HKS Conjecture must be restated : Hof, Knill and Simon Conjecture Conjecture Main Results Let w be a fixed point of a primitive morphism. Then, Further work | Pal ( w ) | = ∞ if and only if there exists a morphism ϕ such that ϕ ( w ) = w and such that ϕ has a conjugate in class P . This question was solved recently in the binary case (B. Tan, 2007).
Main Results Combinatorial properties of f -palindromes Sébastien First, we obtained a result less general then B. Tan : Labbé Aims of the Theorem talk Let Σ = { a , b } , ϕ : Σ ∗ �→ Σ ∗ be a primitive uniform Definitions and notations morphism and w = ϕ ( w ) an fixed point. Then, w contains Hof, Knill and ϕ or ϕ 2 is in arbitrarily long palindromes if and only if ϕ , � Simon Conjecture class P . Main Results Further work Our approach is making use of f -palindromes. Therefore, we also obtained an interesting and similar result for f -palindromes...
Main Results Combinatorial properties of Let f : Σ → Σ be an involution which extends to a morphism f -palindromes on Σ ∗ . We say that w ∈ Σ ∗ is an f -palindrome if w = f ( � w ) . Sébastien Labbé They are also called f -pseudo-palindrome in the litterature Aims of the talk (Anne, Zamboni, Zorca, 2005 ; de Luca, De Luca, 2006 ; Definitions and notations Halava, Harju, Kärki, Zamboni, 2007). Hof, Knill and Simon Example Conjecture Let Σ = { a , b } and E be the involution a �→ b , b �→ a . The Main Results Further work words ε, ab , ba , abab , aabb , baba , bbaa , abbaab , bababa are E -palindromes.
Main Results Combinatorial properties of f -palindromes Definition Sébastien Labbé We say that a morphism ϕ is in class f - P if there exists an Aims of the f -palindrome p and for each α ∈ Σ there exists a talk f -palindrome q α such that ϕ ( α ) = pq α . Definitions and notations Hof, Knill and Our second result is : Simon Conjecture Theorem Main Results Let Σ = { a , b } , ϕ : Σ ∗ �→ Σ ∗ be a primitive uniform Further work morphism and w = ϕ ( w ) an fixed point. If w contains arbitrarily long E-palindromes, then either ϕ , � ϕ , ϕ ◦ µ or ϕ ◦ µ is in class E- P , where µ is the Thue-Morse morphism. �
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