PALINDROMES IN PURE MORPHIC WORDS

Michelangelo Bucci with

Elise Vaslet

a –––› abb –––› a

aababa

abaababaababa

...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 ⟼ aabcacbab ⟼ aac ⟼ a

u = aabcacbaaabcacbaaaaaabcacbaaa...

a ⟼ aabcacba = aPb ⟼ aac ⟼ a

u = aPaPaaaaPaaaaP...

P ...

Pa ...

Pa ...

Pa ...

a ...

PP P

PP Pa

a

u = aPaPaaaaPaaaaP...

w ww w...

u = aPaPaaaaPaaaaP...

w ww w...aP... aP... aP... aP...

u = aPaPaaaaPaaaaP...

w ww w...aP... aP... aP... aP...

u = aPaPaaaaPaaaaP...

w ww w...aP... aP... aP... aP...

w’

u = aPaPaaaaPaaaaP...

w ww w...aP... aP... aP... aP...

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

vπ = lps(v)

vπ = lps(v)

v’

vπ = lps(v)

v’

v’’

vπ = lps(v)

v’

v’’

aabca ⟼ aabcabcaaabcacbaaaaaabcacba

u ∊ a{Pa,Paaaa}*

|v| > K ⇒ P ∊ Fact(lps(v))

w ⟼ aw’, w~ ⟼ aw’’ ⇒ w’ = w’’~

PP

PP PP

PP PPa a

PP PPa a

PP PPa a

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 ⟼ aabcacbab ⟼ aac ⟼ a

a ⟼ aabcacbab ⟼ aac ⟼ a

a ⟼ aabcacbab ⟼ aac ⟼ a

aabca ⟼ aabcabcaaabcacbaaaaaabcacba

Thank you