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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