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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, AN set of infinite words |w| = n length of finite word w = w1 · · · wn

• n set A∗ operation ∼ for w = w1 · · · wn gives

w = wn · · · w1, w is reversal of w palindrome is word such that w = w L(u) set of all factors of an infinite word u Pal(u) set of all palindromes in L(u) left degree degL(w) of w in u is number of letters a ∈ A such that aw is factor of u (likewise right degree degR(w)) w is left special factor if degL(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 limn→∞ ϕn(a) if there exist k ∈ N such that for every pair a, b ∈ A the word ϕk(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 Tβ : [0, 1] → [0, 1) Tβ(x) := βx − ⌊βx⌋ Sequence of non-negative integers (tn)n≥1 defined by ti = ⌊βT i−1

β

(1)⌋ is called Rényi expansion of 1, denoted dβ(1) = t1t2 · · · Theorem (Bertrand) If β is a Pisot number then dβ(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 Tβ : [0, 1] → [0, 1) Tβ(x) := βx − ⌊βx⌋ Sequence of non-negative integers (tn)n≥1 defined by ti = ⌊βT i−1

β

(1)⌋ is called Rényi expansion of 1, denoted dβ(1) = t1t2 · · · Theorem (Bertrand) If β is a Pisot number then dβ(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 Tβ : [0, 1] → [0, 1) Tβ(x) := βx − ⌊βx⌋ Sequence of non-negative integers (tn)n≥1 defined by ti = ⌊βT i−1

β

(1)⌋ is called Rényi expansion of 1, denoted dβ(1) = t1t2 · · · Theorem (Bertrand) If β is a Pisot number then dβ(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 dβ(1) is called a Parry number. β > 1 with finite dβ(1) is called a simple Parry number. Canonical substitution ϕ [Fabre] Simple Parry number β > 1 with dβ(1) = t1 · · · tm ϕ(0) = 0t11 . . . ϕ(m − 2) = 0tm−1(m − 1) ϕ(m − 1) = 0tm Alphabet A = {0, . . . , m − 1}, unique fixed point uβ:= limn→∞ ϕn(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 dβ(1) is called a Parry number. β > 1 with finite dβ(1) is called a simple Parry number. Canonical substitution ϕ [Fabre] Simple Parry number β > 1 with dβ(1) = t1 · · · tm ϕ(0) = 0t11 . . . ϕ(m − 2) = 0tm−1(m − 1) ϕ(m − 1) = 0tm Alphabet A = {0, . . . , m − 1}, unique fixed point uβ:= limn→∞ ϕn(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 dβ(1) is called a Parry number. β > 1 with finite dβ(1) is called a simple Parry number. Canonical substitution ϕ [Fabre] Simple Parry number β > 1 with dβ(1) = t1 · · · tm ϕ(0) = 0t11 . . . ϕ(m − 2) = 0tm−1(m − 1) ϕ(m − 1) = 0tm Alphabet A = {0, . . . , m − 1}, unique fixed point uβ:= limn→∞ ϕn(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. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 0010010000100100001001 · · ·

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20

Beta-numeration system

Canonical substitution — Example

• Example. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 0010010000100100001001 · · ·

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20

Beta-numeration system

Canonical substitution — Example

• Example. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 0010010000100100001001 · · ·

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20

Beta-numeration system

Canonical substitution — Example

• Example. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 0010010000100100001001 · · ·

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20

Beta-numeration system

Canonical substitution — Example

• Example. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 0010010000100100001001 · · ·

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20

Beta-numeration system

Canonical substitution — Example

• Example. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 0010010000100100001001 · · ·

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 6 / 20

Beta-numeration system

Canonical substitution — Example

• Example. x2 − 2x − 2, dβ(1) = 22

Alphabet A = {0, 1, . . . , m − 1} = {0, 1} Canonical substitution ϕ(0) = 0t11 = 001 , ϕ(1) = 0t2 = 00 . Fixed point uβ = limn→∞ ϕn(0) uβ = 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 Cu : N → N Cu(n) := #{w | w ∈ L(u), |w| = n} . Theorem (Morse, Hedlund) If there exist n such that Cu(n) ≤ n then u is eventually periodic. Sturmian word is an infinite word u such that Cu(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 w1 and one right special factor w2, and degL(w1) = degR(w2) = 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 Cu : N → N Cu(n) := #{w | w ∈ L(u), |w| = n} . Theorem (Morse, Hedlund) If there exist n such that Cu(n) ≤ n then u is eventually periodic. Sturmian word is an infinite word u such that Cu(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 w1 and one right special factor w2, and degL(w1) = degR(w2) = 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 Cu : N → N Cu(n) := #{w | w ∈ L(u), |w| = n} . Theorem (Morse, Hedlund) If there exist n such that Cu(n) ≤ n then u is eventually periodic. Sturmian word is an infinite word u such that Cu(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 w1 and one right special factor w2, and degL(w1) = degR(w2) = m. Complexity is (m − 1)n + 1.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 7 / 20

Palindromic complexity

Palindromic complexity is a function counting number of different palindroms in an infinite word u Pu : N → N Pu(n) := #{w | w ∈ Pal(u), |w| = n} If for p ∈ Pal(u) there exists a ∈ A such that apa ∈ Pal(u), then apa is palindromic extension of p A palindrome p ∈ Pal(uβ) wich has no palindromic extension is called a maximal palindrome Theorem (Damanik, Zamboni) In an Arnoux-Rauzy word u every palindrome has exactly one palindromic extension. Therefore Pu(2n) = Pu(0) = 1 and Pu(2n + 1) = Pu(1) = #A

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 8 / 20

Palindromic complexity

Palindromic complexity is a function counting number of different palindroms in an infinite word u Pu : N → N Pu(n) := #{w | w ∈ Pal(u), |w| = n} If for p ∈ Pal(u) there exists a ∈ A such that apa ∈ Pal(u), then apa is palindromic extension of p A palindrome p ∈ Pal(uβ) wich has no palindromic extension is called a maximal palindrome Theorem (Damanik, Zamboni) In an Arnoux-Rauzy word u every palindrome has exactly one palindromic extension. Therefore Pu(2n) = Pu(0) = 1 and Pu(2n + 1) = Pu(1) = #A

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 8 / 20

Palindromic complexity

Palindromic complexity is a function counting number of different palindroms in an infinite word u Pu : N → N Pu(n) := #{w | w ∈ Pal(u), |w| = n} If for p ∈ Pal(u) there exists a ∈ A such that apa ∈ Pal(u), then apa is palindromic extension of p A palindrome p ∈ Pal(uβ) wich has no palindromic extension is called a maximal palindrome Theorem (Damanik, Zamboni) In an Arnoux-Rauzy word u every palindrome has exactly one palindromic extension. Therefore Pu(2n) = Pu(0) = 1 and Pu(2n + 1) = Pu(1) = #A

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 8 / 20

Palindromic complexity

Palindromic complexity is a function counting number of different palindroms in an infinite word u Pu : N → N Pu(n) := #{w | w ∈ Pal(u), |w| = n} If for p ∈ Pal(u) there exists a ∈ A such that apa ∈ Pal(u), then apa is palindromic extension of p A palindrome p ∈ Pal(uβ) wich has no palindromic extension is called a maximal palindrome Theorem (Damanik, Zamboni) In an Arnoux-Rauzy word u every palindrome has exactly one palindromic extension. Therefore Pu(2n) = Pu(0) = 1 and Pu(2n + 1) = Pu(1) = #A

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 8 / 20

Palindromic complexity

Words with bounded number of palindromes

An infinite word u is uniformly recurrent if every factor in L(u) occurs in u infinitely many times, and with bounded gaps Lemma If the language L(u) of a uniformly recurrent word u contains infinitely many palindromes, then L(u) is closed under reversal.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 9 / 20

Palindromic complexity

Words with bounded number of palindromes

An infinite word u is uniformly recurrent if every factor in L(u) occurs in u infinitely many times, and with bounded gaps Lemma If the language L(u) of a uniformly recurrent word u contains infinitely many palindromes, then L(u) is closed under reversal.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 9 / 20

Palindromic complexity

Words with bounded number of palindromes

An infinite word u is uniformly recurrent if every factor in L(u) occurs in u infinitely many times, and with bounded gaps Lemma If the language L(u) of a uniformly recurrent word u contains infinitely many palindromes, then L(u) is closed under reversal. Proof for all n there exits R(n) such that arbitrary factor of u of length R(n) contains all factors of u of length n Since #Pal(u) = ∞ it must contain palindrome p such that |p| > R(n) p contains all factors of u of length n and p is palindrome hence with each factor it contains also its reversal

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 9 / 20

Palindromic complexity

Words with bounded number of palindromes

An infinite word u is uniformly recurrent if every factor in L(u) occurs in u infinitely many times, and with bounded gaps Lemma If the language L(u) of a uniformly recurrent word u contains infinitely many palindromes, then L(u) is closed under reversal. Proof for all n there exits R(n) such that arbitrary factor of u of length R(n) contains all factors of u of length n Since #Pal(u) = ∞ it must contain palindrome p such that |p| > R(n) p contains all factors of u of length n and p is palindrome hence with each factor it contains also its reversal

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 9 / 20

Palindromic complexity

Words with bounded number of palindromes

An infinite word u is uniformly recurrent if every factor in L(u) occurs in u infinitely many times, and with bounded gaps Lemma If the language L(u) of a uniformly recurrent word u contains infinitely many palindromes, then L(u) is closed under reversal. Proof for all n there exits R(n) such that arbitrary factor of u of length R(n) contains all factors of u of length n Since #Pal(u) = ∞ it must contain palindrome p such that |p| > R(n) p contains all factors of u of length n and p is palindrome hence with each factor it contains also its reversal

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 9 / 20

Palindromic complexity

Words with bounded number of palindromes

An infinite word u is uniformly recurrent if every factor in L(u) occurs in u infinitely many times, and with bounded gaps Lemma If the language L(u) of a uniformly recurrent word u contains infinitely many palindromes, then L(u) is closed under reversal. Remark. Fixed point of primitive substitution is uniformly recurrent uβ is fixed point of primitive substitution

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 9 / 20

Palindromic complexity

Condition (C)

Proposition (Frougny, Masáková, Pelnatová) Let β > 1 be a simple Parry number with dβ(1) = t1 · · · tm. i) L(uβ) is closed under reversal if and only if Condition (C): t1 = t2 = · · · = tm−1 ii) uβ is Arnoux-Rauzy iff (C) is satisfied and tm = 1 Corollary Let β be a simple Parry number which does not satisfy condition (C). Then there exists n0 ∈ N such that Puβ(n) = 0 for n ≥ n0.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 10 / 20

Palindromic complexity

Condition (C)

Proposition (Frougny, Masáková, Pelnatová) Let β > 1 be a simple Parry number with dβ(1) = t1 · · · tm. i) L(uβ) is closed under reversal if and only if Condition (C): t1 = t2 = · · · = tm−1 ii) uβ is Arnoux-Rauzy iff (C) is satisfied and tm = 1 Corollary Let β be a simple Parry number which does not satisfy condition (C). Then there exists n0 ∈ N such that Puβ(n) = 0 for n ≥ n0.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 10 / 20

Palindromic complexity

Confluent Pisot numbers

We consider only Pisot numbers satisfying (C), called confluent t := t1 = t2 = · · · = tm−1 and s := tm Canonical substitution ϕ ϕ(0) = 0t1 ϕ(1) = 0t2 . . . ϕ(m − 2) = 0t(m − 1) ϕ(m − 1) = 0s

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 11 / 20

Palindromic complexity

Confluent Pisot numbers

We consider only Pisot numbers satisfying (C), called confluent t := t1 = t2 = · · · = tm−1 and s := tm Canonical substitution ϕ ϕ(0) = 0t1 ϕ(1) = 0t2 . . . ϕ(m − 2) = 0t(m − 1) ϕ(m − 1) = 0s

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 11 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Words U(n) are maximal left special factors of uβ Every matimal left special factor v of uβ is of the form U(n) A word V (n) is the longest common prefix of uβ and U(n) Words V (n) are total bispecial factors of uβ

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 12 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Words U(n) are maximal left special factors of uβ Every matimal left special factor v of uβ is of the form U(n) A word V (n) is the longest common prefix of uβ and U(n) Words V (n) are total bispecial factors of uβ

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 12 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Words U(n) are maximal left special factors of uβ Every matimal left special factor v of uβ is of the form U(n) A word V (n) is the longest common prefix of uβ and U(n) Words V (n) are total bispecial factors of uβ

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 12 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Words U(n) are maximal left special factors of uβ Every matimal left special factor v of uβ is of the form U(n) A word V (n) is the longest common prefix of uβ and U(n) Words V (n) are total bispecial factors of uβ

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 12 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Words U(n) are maximal left special factors of uβ Every matimal left special factor v of uβ is of the form U(n) A word V (n) is the longest common prefix of uβ and U(n) Words V (n) are total bispecial factors of uβ

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 12 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 U(1) = 03 = 000 , U(2) = ϕ(U(1))02 = 00100100100 , U(3) = ϕ(U(2))02 = 00100100001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 V (1) = 02 = 00 , V (2) = ϕ(V (1))02 = 00100100 , V (3) = ϕ(V (2))02 = 001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 V (1) = 02 = 00 , V (2) = ϕ(V (1))02 = 00100100 , V (3) = ϕ(V (2))02 = 001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 V (1) = 02 = 00 , V (2) = ϕ(V (1))02 = 00100100 , V (3) = ϕ(V (2))02 = 001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 V (1) = 02 = 00 , V (2) = ϕ(V (1))02 = 00100100 , V (3) = ϕ(V (2))02 = 001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 V (1) = 02 = 00 , V (2) = ϕ(V (1))02 = 00100100 , V (3) = ϕ(V (2))02 = 001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n) — Example

• Example. dβ(1) = 22, t = 2, s = 2, ϕ(0) = 001, ϕ(1) = 00

ε-0-0 1-0-0-1-0-0

• 0-0-1-0-0-1-0-0-0-0-1-0-0-1-0-0
• 1-0-0

1-0-0-1-0-0-0-0-1· · · 0-0-1-0-0-1-0-0 V (1) = 02 = 00 , V (2) = ϕ(V (1))02 = 00100100 , V (3) = ϕ(V (2))02 = 001001000010010000100100 .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 13 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Several properties, used mainly in obtaining of Cuβ ∆C(n) =

• m

if |V (k)| < n ≤ |U(k)| for some k ∈ N , m − 1

• therwise .
• r

∆2C(n) = ∆C(n + 1) − ∆C(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 14 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Several properties, used mainly in obtaining of Cuβ ∆C(n) =

• m

if |V (k)| < n ≤ |U(k)| for some k ∈ N , m − 1

• therwise .
• r

∆2C(n) = ∆C(n + 1) − ∆C(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 14 / 20

Palindromic complexity

Words U(n) and V (n)

Sequences of words U(n) and V (n) U(1) := 0t+s−1, U(n) := ϕ(U(n−1))0t for n ≥ 2 . V (1) := 0t, V (n) := ϕ(V (n−1))0t for n ≥ 2 . Several properties, used mainly in obtaining of Cuβ ∆C(n) =

• m

if |V (k)| < n ≤ |U(k)| for some k ∈ N , m − 1

• therwise .
• r

∆2C(n) = ∆C(n + 1) − ∆C(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 14 / 20

Palindromic complexity

Non Arnoux-Rauzy case

Proposition Let uβ be the fixed point of the substitution ϕ with parameters t ≥ s ≥ 2, and let p be a palindrome in uβ. Then (i) p is a maximal palindrome if and only if p = U(n) for some n ∈ N; (ii) p has two palindromic extensions in uβ if and only if p = V (n) for some n ∈ N; (iii) p has a unique palindromic extension if and only if p = U(n), p = V (n) for all n ∈ N. Or equivalently P(n + 2) − P(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 15 / 20

Palindromic complexity

Non Arnoux-Rauzy case

Proposition Let uβ be the fixed point of the substitution ϕ with parameters t ≥ s ≥ 2, and let p be a palindrome in uβ. Then (i) p is a maximal palindrome if and only if p = U(n) for some n ∈ N; (ii) p has two palindromic extensions in uβ if and only if p = V (n) for some n ∈ N; (iii) p has a unique palindromic extension if and only if p = U(n), p = V (n) for all n ∈ N. Or equivalently P(n + 2) − P(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 15 / 20

Palindromic complexity

Non Arnoux-Rauzy case

Recall that ∆C(n + 1) − ∆C(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P(n + 2) − P(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

Corollary We have for s ≥ 2, for all n ∈ N P(n + 2) − P(n) = ∆C(n + 1) − ∆C(n) .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 16 / 20

Palindromic complexity

Non Arnoux-Rauzy case

Recall that ∆C(n + 1) − ∆C(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

P(n + 2) − P(n) =    1 if n = |V (k)| , −1 if n = |U(k)| ,

• therwise .

Corollary We have for s ≥ 2, for all n ∈ N P(n + 2) − P(n) = ∆C(n + 1) − ∆C(n) .

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 16 / 20

Palindromic complexity

Connection between factor and palindromic complexity

Theorem Let uβ be the fixed point of the substitution ϕ. Then P(n + 1) + P(n) = ∆C(n) + 2 , for n ∈ N . Remark. Recall that ∆C(n) ≤ #A. This implies P(n + 1) + P(n) ≤ #A + 2 , and thus the palindromic complexity is bounded.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 17 / 20

Palindromic complexity

Connection between factor and palindromic complexity

Theorem Let uβ be the fixed point of the substitution ϕ. Then P(n + 1) + P(n) = ∆C(n) + 2 , for n ∈ N . Remark. Recall that ∆C(n) ≤ #A. This implies P(n + 1) + P(n) ≤ #A + 2 , and thus the palindromic complexity is bounded.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 17 / 20

Palindromic complexity

Explicit values

Theorem Let uβ be the fixed point of the substitution ϕ, with parameters t ≥ s ≥ 2. (i) Let s be odd and let t be even. Then P(2n + 1) = m P(2n) =

• 2,

if |V (k)| < 2n ≤ |U(k)| for some k 1,

• therwise.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 18 / 20

Palindromic complexity

Explicit values

Theorem Let uβ be the fixed point of the substitution ϕ, with parameters t ≥ s ≥ 2. (ii) Let s and t be odd. Then P(2n + 1) =      m + 1, if |V (k)| < 2n + 1 ≤ |U(k)| with k ≡ m (mod (m + 1)) m,

• therwise.

P(2n) =      2, if |V (k)| < 2n ≤ |U(k)| with k ≡ m (mod (m + 1)) 1,

• therwise

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 18 / 20

Palindromic complexity

Explicit values

Theorem Let uβ be the fixed point of the substitution ϕ, with parameters t ≥ s ≥ 2. (iii) Let s be even and t be odd. Then P(2n + 1) =        m + 2, if |V (k)| < 2n + 1 ≤ |U(k)| , k ≥ 2 m, if 2n + 1 ≤ |V (1)| m + 1,

• therwise.

P(2n) =

• 1,

if 2n ≤ |U(1)| 0,

• therwise

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 18 / 20

Palindromic complexity

Explicit values

Theorem Let uβ be the fixed point of the substitution ϕ, with parameters t ≥ s ≥ 2. (iv) Let s and t be even. Then P(2n + 1) =

• #
• k ≤ m
• 2n + 1 ≤ |U(k)|
• ,

2n + 1 ≤ |U(m)| 0,

• therwise

P(2n)| =            m + 2, |V (k)| < 2n ≤ |U(k)| k ≥ m + 1 #

• k ≤ m
• 2n ≤ |V (k)|
• ,

2n ≤ |V (m+1)| m + 1,

• therwise.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 18 / 20

Palindromic complexity

Length of words U(n) and V (n)

For the determination of the value P(n) for a given n, we have to know lengths |V (k)|, |U(k)|. Proposition (Frougny, Masáková, Pelantová) |V (k)| = t

k−1

• i=0

Gi , and |U(k)| = |V (k)| + (s − 1)Gk−1 , where Gn is a sequence of integers defined by the recurrence G0 = 1 , Gn = t(Gn−1 + · · · + G0) + 1 , for 1 ≤ n ≤ m − 1 , Gn = t(Gn−1 + · · · + Gn−m+1) + sGn−m , for n ≥ m . [Bertrand]: The sequence (Gn)n∈N defines the canonical linear numeration system associated with the number β.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 19 / 20

Palindromic complexity

Length of words U(n) and V (n)

For the determination of the value P(n) for a given n, we have to know lengths |V (k)|, |U(k)|. Proposition (Frougny, Masáková, Pelantová) |V (k)| = t

k−1

• i=0

Gi , and |U(k)| = |V (k)| + (s − 1)Gk−1 , where Gn is a sequence of integers defined by the recurrence G0 = 1 , Gn = t(Gn−1 + · · · + G0) + 1 , for 1 ≤ n ≤ m − 1 , Gn = t(Gn−1 + · · · + Gn−m+1) + sGn−m , for n ≥ m . [Bertrand]: The sequence (Gn)n∈N defines the canonical linear numeration system associated with the number β.

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 19 / 20

Non-simple Parry numbers

Known results [Bernat]: for non-simple Parry number, uβ is closed under reversal iff β is quadratic irrationality with minimal polynomial x2 − ax + b, where a ≥ b + 2, b ≥ 1. [Balková]: Factor and palindromic complexity for this class of Parry numbers

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 20 / 20

Non-simple Parry numbers

Known results [Bernat]: for non-simple Parry number, uβ is closed under reversal iff β is quadratic irrationality with minimal polynomial x2 − ax + b, where a ≥ b + 2, b ≥ 1. [Balková]: Factor and palindromic complexity for this class of Parry numbers

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 20 / 20

Non-simple Parry numbers

Known results [Bernat]: for non-simple Parry number, uβ is closed under reversal iff β is quadratic irrationality with minimal polynomial x2 − ax + b, where a ≥ b + 2, b ≥ 1. [Balková]: Factor and palindromic complexity for this class of Parry numbers

P . Ambrož (FNSPE & LIAFA) Palindromic complexity of simple Parry . . . 20 / 20