Palindromic complexity of infinite words associated with simple Parry numbers P . Ambrož joint work with C. Frougny, Z. Masáková, E. Pelantová Seminᡠr kombinatorické a algebraické struktury, 21. dubna 2006 P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 1 / 20
Notation Words A alphabet, A ∗ free monoid, ε empty word, A N set of infinite words | w | = n length of finite word w = w 1 · · · w n on set A ∗ operation ∼ for w = w 1 · · · w n gives � w = w n · · · w 1 , � w is reversal of w palindrome is word such that w = � w L ( u ) set of all factors of an infinite word u P al ( u ) set of all palindromes in L ( u ) left degree deg L ( w ) of w in u is number of letters a ∈ A such that aw is factor of u (likewise right degree deg R ( w ) ) w is left special factor if deg L ( w ) ≥ 2 (likewise right special factor) P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 2 / 20
Notation Substitutions morphism ϕ is mapping on A ∗ such that ϕ ( vw ) = ϕ ( v ) ϕ ( w ) for all v , w ∈ A ∗ non-erasing morphism: ϕ ( a ) � = ε for all a ∈ A non-erasing morphism such that ϕ ( a ) = aw for some a ∈ A and non-empty w ∈ A ∗ is called substitution an infinite word u is fixed point of ϕ if u = ϕ ( u ) substitution has at least one fixed point lim n →∞ ϕ n ( a ) if there exist k ∈ N such that for every pair a , b ∈ A the word ϕ k ( a ) contains letter b , ϕ is called primitive P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 3 / 20
Beta-numeration system Beta-transformation For real number β > 1, β -transformation T β : [ 0 , 1 ] �→ [ 0 , 1 ) T β ( x ) := β x − ⌊ β x ⌋ Sequence of non-negative integers ( t n ) n ≥ 1 defined by t i = ⌊ β T i − 1 ( 1 ) ⌋ β is called Rényi expansion of 1, denoted d β ( 1 ) = t 1 t 2 · · · Theorem (Bertrand) If β is a Pisot number then d β ( 1 ) is eventuelly periodic. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 4 / 20
Beta-numeration system Beta-transformation For real number β > 1, β -transformation T β : [ 0 , 1 ] �→ [ 0 , 1 ) T β ( x ) := β x − ⌊ β x ⌋ Sequence of non-negative integers ( t n ) n ≥ 1 defined by t i = ⌊ β T i − 1 ( 1 ) ⌋ β is called Rényi expansion of 1, denoted d β ( 1 ) = t 1 t 2 · · · Theorem (Bertrand) If β is a Pisot number then d β ( 1 ) is eventuelly periodic. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 4 / 20
Beta-numeration system Beta-transformation For real number β > 1, β -transformation T β : [ 0 , 1 ] �→ [ 0 , 1 ) T β ( x ) := β x − ⌊ β x ⌋ Sequence of non-negative integers ( t n ) n ≥ 1 defined by t i = ⌊ β T i − 1 ( 1 ) ⌋ β is called Rényi expansion of 1, denoted d β ( 1 ) = t 1 t 2 · · · Theorem (Bertrand) If β is a Pisot number then d β ( 1 ) is eventuelly periodic. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 4 / 20
Beta-numeration system Canonical substitution β > 1 with eventually periodic d β ( 1 ) is called a Parry number. β > 1 with finite d β ( 1 ) is called a simple Parry number. Canonical substitution ϕ [ Fabre ] Simple Parry number β > 1 with d β ( 1 ) = t 1 · · · t m 0 t 1 1 ϕ ( 0 ) = . . . 0 t m − 1 ( m − 1 ) ϕ ( m − 2 ) = 0 t m ϕ ( m − 1 ) = Alphabet A = { 0 , . . . , m − 1 } , unique fixed point u β := lim n →∞ ϕ n ( 0 ) [ Canterini, Siegel ] : Substitution ϕ is primitive. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 5 / 20
Beta-numeration system Canonical substitution β > 1 with eventually periodic d β ( 1 ) is called a Parry number. β > 1 with finite d β ( 1 ) is called a simple Parry number. Canonical substitution ϕ [ Fabre ] Simple Parry number β > 1 with d β ( 1 ) = t 1 · · · t m 0 t 1 1 ϕ ( 0 ) = . . . 0 t m − 1 ( m − 1 ) ϕ ( m − 2 ) = 0 t m ϕ ( m − 1 ) = Alphabet A = { 0 , . . . , m − 1 } , unique fixed point u β := lim n →∞ ϕ n ( 0 ) [ Canterini, Siegel ] : Substitution ϕ is primitive. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 5 / 20
Beta-numeration system Canonical substitution β > 1 with eventually periodic d β ( 1 ) is called a Parry number. β > 1 with finite d β ( 1 ) is called a simple Parry number. Canonical substitution ϕ [ Fabre ] Simple Parry number β > 1 with d β ( 1 ) = t 1 · · · t m 0 t 1 1 ϕ ( 0 ) = . . . 0 t m − 1 ( m − 1 ) ϕ ( m − 2 ) = 0 t m ϕ ( m − 1 ) = Alphabet A = { 0 , . . . , m − 1 } , unique fixed point u β := lim n →∞ ϕ n ( 0 ) [ Canterini, Siegel ] : Substitution ϕ is primitive. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 5 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Beta-numeration system Canonical substitution — Example Example. x 2 − 2 x − 2, d β ( 1 ) = 22 Alphabet A = { 0 , 1 , . . . , m − 1 } = { 0 , 1 } Canonical substitution ϕ ( 0 ) = 0 t 1 1 = 001 , ϕ ( 1 ) = 0 t 2 = 00 . Fixed point u β = lim n →∞ ϕ n ( 0 ) u β = 0010010000100100001001 · · · P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20
Subword complexity Subword complexity is function counting number of different factors of an infinite word u C u : N → N C u ( n ) := # { w | w ∈ L ( u ) , | w | = n } . Theorem (Morse, Hedlund) If there exist n such that C u ( n ) ≤ n then u is eventually periodic. Sturmian word is an infinite word u such that C u ( n ) = n + 1 Arnoux-Rauzy word of order m – over m letter alphabet such that for each n there exist exactly one left special factor w 1 and one right special factor w 2 , and deg L ( w 1 ) = deg R ( w 2 ) = m . Complexity is ( m − 1 ) n + 1. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 7 / 20
Subword complexity Subword complexity is function counting number of different factors of an infinite word u C u : N → N C u ( n ) := # { w | w ∈ L ( u ) , | w | = n } . Theorem (Morse, Hedlund) If there exist n such that C u ( n ) ≤ n then u is eventually periodic. Sturmian word is an infinite word u such that C u ( n ) = n + 1 Arnoux-Rauzy word of order m – over m letter alphabet such that for each n there exist exactly one left special factor w 1 and one right special factor w 2 , and deg L ( w 1 ) = deg R ( w 2 ) = m . Complexity is ( m − 1 ) n + 1. P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 7 / 20
Recommend
More recommend
