PALINDROMES IN PURE MORPHIC WORDS Michelangelo Bucci with Elise - PowerPoint PPT Presentation

PALINDROMES IN PURE MORPHIC WORDS Michelangelo Bucci with Elise Vaslet

a –––› ab b –––› a a ab aba abaab abaababa ... abaababaabaababaababaaba...

beeblebrox ~ =xorbelbeeb Pal = {w | w ~ = w} Pal(w) = { v ∈ Fact(w) s. t. v ∈ Pal } P(n) = #{ v ∈ Pal(w) s. t. |v| = n }

#Pal(v) ≤ |v| + 1

D(w) = |w| + 1 - #Pal(w)

abca aababbaa PwP s.t. w ~ ≠ w

Palindromic defect conjecture (Blondin Massé, Brlek, Garon, Labbé): Let u be the fixed point of a primitive morphism. If 0<D(u)< ∞ then u is periodic.

Palindromic defect conjecture (Blondin Massé, Brlek, Garon, Labbé): Let u be the fixed point of a primitive morphism. If 0<D(u)< ∞ then u is periodic.

Palindromic defect conjecture (Blondin Massé, Brlek, Garon, Labbé): Let u be the fixed point of a primitive morphism. If 0<D(u)< ∞ then u is periodic. BUT...

a ⟼ aabcacba b ⟼ aa c ⟼ a u = aabcacbaaabcacbaaaaaabcacbaaa...

a ⟼ aabcacba = aP b ⟼ aa c ⟼ a u = aPaPaaaaPaaaaP ...

P ...

a P ...

a P ...

a P ... a ...

P P P

a P P P a

u = aPaPaaaaPaaaaP ... ... w w w w

u = aPaPaaaaPaaaaP ... aP ... aP ... aP ... aP ... ... w w w w

u = aPaPaaaaPaaaaP ... aP ... aP ... aP ... aP ... ... w w w w

u = aPaPaaaaPaaaaP ... aP ... aP ... aP ... aP ... ... w w w w w’

u = aPaPaaaaPaaaaP ... aP ... aP ... aP ... aP ... ... w w w w ... w’ w’ w’ w’

D(u) = #{ v s.t. v ∊ Pref(u) and v is end-lacunary }

D(u) = #{ v s.t. v ∊ Pref(u) and v is end-lacunary } u = aabca...

D(u) = #{ v s.t. v ∊ Pref(u) and v is end-lacunary } u = aabca... D(u) ≥ 1

v π = lps(v)

v π = lps(v) v’

v π = lps(v) v’ v’’

v π = lps(v) v’ v’’ aabca ⟼ aabcabcaaabcacbaaaaaabcacba

u ∊ a{Pa,Paaaa} * |v| > K ⇒ P ∊ Fact(lps(v)) w ⟼ aw’, w ~ ⟼ aw’’ ⇒ w’ = w’’ ~

P P

P P P P

a P P a P P

a P P a P P

a P P a P P

P

P

aaa

aaa P

Hence (?) u is an aperiodic fixed point of a primitive morphism and D(u)=1

Hence (?) u is an aperiodic fixed point of a primitive morphism and D(u)=1 BUT...

a ⟼ aabcacba b ⟼ aa c ⟼ a

a ⟼ aabcacba b ⟼ aa c ⟼ a

a ⟼ aabcacba b ⟼ aa c ⟼ a

aabca ⟼ aabcabcaaabcacbaaaaaabcacba

Thank you