SLIDE 1
PALINDROMES IN PURE MORPHIC WORDS Michelangelo Bucci with Elise - - PowerPoint PPT Presentation
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.
SLIDE 2
SLIDE 3
beeblebrox~=xorbelbeeb Pal = {w | w~ = w} Pal(w) = { v ∈ Fact(w) s. t. v ∈ Pal } P(n) = #{ v ∈ Pal(w) s. t. |v| = n }
SLIDE 4
#Pal(v) ≤ |v| + 1
SLIDE 5
D(w) = |w| + 1 - #Pal(w)
SLIDE 6
abca aababbaa PwP s.t. w~≠w
SLIDE 7
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.
SLIDE 8
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.
SLIDE 9
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...
SLIDE 10
a ⟼ aabcacba b ⟼ aa c ⟼ a u = aabcacbaaabcacbaaaaaabcacbaaa...
SLIDE 11
a ⟼ aabcacba = aP b ⟼ aa c ⟼ a u = aPaPaaaaPaaaaP ...
SLIDE 12
P ...
SLIDE 13
P a ...
SLIDE 14
P a ...
SLIDE 15
P a ... a ...
SLIDE 16
P P P
SLIDE 17
P P P a a
SLIDE 18
u = aPaPaaaaPaaaaP ... w w w w ...
SLIDE 19
u = aPaPaaaaPaaaaP ... w w w w ... aP ... aP ... aP ... aP ...
SLIDE 20
u = aPaPaaaaPaaaaP ... w w w w ... aP ... aP ... aP ... aP ...
SLIDE 21
u = aPaPaaaaPaaaaP ... w w w w ... aP ... aP ... aP ... aP ... w’
SLIDE 22
u = aPaPaaaaPaaaaP ... w w w w ... aP ... aP ... aP ... aP ... w’ w’ w’ ... w’
SLIDE 23
D(u) = #{ v s.t. v∊Pref(u) and v is end-lacunary }
SLIDE 24
D(u) = #{ v s.t. v∊Pref(u) and v is end-lacunary } u = aabca...
SLIDE 25
D(u) = #{ v s.t. v∊Pref(u) and v is end-lacunary } u = aabca... D(u) ≥ 1
SLIDE 26
v π = lps(v)
SLIDE 27
v π = lps(v) v’
SLIDE 28
v π = lps(v) v’ v’’
SLIDE 29
v π = lps(v) v’ v’’ aabca ⟼ aabcabcaaabcacbaaaaaabcacba
SLIDE 30
u ∊ a{Pa,Paaaa}* |v| > K ⇒ P ∊ Fact(lps(v)) w ⟼ aw’, w~ ⟼ aw’’ ⇒ w’ = w’’~
SLIDE 31
SLIDE 32
P P
SLIDE 33
P P P P
SLIDE 34
P P P P a a
SLIDE 35
P P P P a a
SLIDE 36
P P P P a a
SLIDE 37
SLIDE 38
P
SLIDE 39
P
SLIDE 40
aaa
SLIDE 41
aaa P
SLIDE 42
Hence (?) u is an aperiodic fixed point of a primitive morphism and D(u)=1
SLIDE 43
Hence (?) u is an aperiodic fixed point of a primitive morphism and D(u)=1
BUT...
SLIDE 44
a ⟼ aabcacba b ⟼ aa c ⟼ a
SLIDE 45
a ⟼ aabcacba b ⟼ aa c ⟼ a
SLIDE 46
a ⟼ aabcacba b ⟼ aa c ⟼ a
SLIDE 47
SLIDE 48
aabca ⟼ aabcabcaaabcacbaaaaaabcacba
SLIDE 49