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Rich, Sturmian & trapezoidal words Amy Glen School of Chemical & Mathematical Sciences Murdoch University, Perth, Australia amy.glen@gmail.com http://wwwstaff.murdoch.edu.au/~aglen 54th AustMS Annual Meeting @ The University of

  1. Finite and Infinite Words Words . . . Depending on the problem to be solved, it may be fruitful to study combinatorial and structural properties of the words representing the elements of a particular set or to impose certain combinatorial conditions on such words. Most commonly studied words are those which satisfy one or more strong regularity properties; for instance, words containing many repetitions or palindromes. The extent to which a word exhibits strong regularity properties is generally inversely proportional to its “complexity”. Basic measure: number of distinct blocks (factors) of each length occurring in the word. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 4 / 40

  2. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  3. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  4. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  5. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  6. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 F 2 ( x ) = { 00 , 01 , 10 , 11 } , C x ( 2 ) = 4 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  7. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 F 2 ( x ) = { 00 , 01 , 10 , 11 } , C x ( 2 ) = 4 F 3 ( x ) = { 000 , 001 , 010 , 100 , 101 , 110 , 111 } , C x ( 3 ) = 8 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  8. Finite and Infinite Words Factor Complexity Given a finite or infinite word w , let F n ( w ) denote the set of distinct factors of w of length n ∈ N + . The function C w ( n ) : N → N defined by C w ( n ) = Card ( F n ( w )) is called the factor complexity function of w . Example √ x = ( 2 ) 2 = 1 . 0110101000001001111 . . . F 1 ( x ) = { 0 , 1 } , C x ( 1 ) = 2 F 2 ( x ) = { 00 , 01 , 10 , 11 } , C x ( 2 ) = 4 F 3 ( x ) = { 000 , 001 , 010 , 100 , 101 , 110 , 111 } , C x ( 3 ) = 8 √ Conjecture: C x ( n ) = 2 n for all n as it is believed 2 is normal in base 2. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 5 / 40

  9. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  10. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  11. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  12. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Low complexity accounts for many interesting features, as it induces certain regularities in such words without, however, making them periodic. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  13. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Low complexity accounts for many interesting features, as it induces certain regularities in such words without, however, making them periodic. References in: Combinatorics, Symbolic Dynamics, Number Theory, Discrete Geometry, Theoretical Physics, Theoretical Computer Science. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  14. Complexity & Sturmian words Complexity & Periodicity Theorem (Morse-Hedlund 1940) An infinite word w is eventually periodic if and only if C w ( n ) ≤ n for some n ∈ N + . That is: w is aperiodic ⇔ C w ( n ) ≥ n + 1 for all n ∈ N . Sturmian words are the aperiodic infinite words of minimal complexity, i.e., an infinite word w is Sturmian if and only if C w ( n ) = n + 1 for each n . Low complexity accounts for many interesting features, as it induces certain regularities in such words without, however, making them periodic. References in: Combinatorics, Symbolic Dynamics, Number Theory, Discrete Geometry, Theoretical Physics, Theoretical Computer Science. Numerous equivalent definitions & characterisations . . . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 6 / 40

  15. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  16. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Theorem (Droubay-Pirillo 1999) An infinite word w is Sturmian if and only if � 1 if n is even P w ( n ) = 2 if n is odd Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  17. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Theorem (Droubay-Pirillo 1999) An infinite word w is Sturmian if and only if � 1 if n is even P w ( n ) = 2 if n is odd Note: Any Sturmian word is over a 2-letter alphabet since it has two distinct factors of length 1. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  18. Complexity & Sturmian words Palindrome Complexity Given a finite or infinite word w , let P w ( n ) denote the palindromic complexity function of w , which counts the number of palindromic factors of w of each length n ≥ 0. Theorem (Droubay-Pirillo 1999) An infinite word w is Sturmian if and only if � 1 if n is even P w ( n ) = 2 if n is odd Note: Any Sturmian word is over a 2-letter alphabet since it has two distinct factors of length 1. A Sturmian word over the alphabet { a , b } contains either aa or bb , but not both. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 7 / 40

  19. Complexity & Sturmian words Constructing Sturmian words Let’s consider a nice geometric realisation, starting with a special class of finite words . . . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 8 / 40

  20. Complexity & Sturmian words Constructing Sturmian words Let’s consider a nice geometric realisation, starting with a special class of finite words . . . Words over a 2-letter alphabet { a , b } that are factors of (infinite) Sturmian words are called finite Sturmian words – they are the cyclic shifts of Christoffel words obtained via the following construction. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 8 / 40

  21. Complexity & Sturmian words Constructing Sturmian words Let’s consider a nice geometric realisation, starting with a special class of finite words . . . Words over a 2-letter alphabet { a , b } that are factors of (infinite) Sturmian words are called finite Sturmian words – they are the cyclic shifts of Christoffel words obtained via the following construction. Using a similar construction we obtain infinite Sturmian words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 8 / 40

  22. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 9 / 40

  23. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 10 / 40

  24. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 11 / 40

  25. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 12 / 40

  26. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 13 / 40

  27. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 14 / 40

  28. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 15 / 40

  29. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 16 / 40

  30. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 17 / 40

  31. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a L(3,5) = a Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 18 / 40

  32. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a a L(3,5) = aa Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 19 / 40

  33. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 b a a L(3,5) = aab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 20 / 40

  34. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a b a a L(3,5) = aaba Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 21 / 40

  35. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a a b a a L(3,5) = aabaa Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 22 / 40

  36. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 b a a b a a L(3,5) = aabaab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 23 / 40

  37. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 a b a a b a a L(3,5) = aabaaba Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 24 / 40

  38. Complexity & Sturmian words Christoffel words: Construction by example Lower Christoffel word of slope 3 5 b a b a a b a a L(3,5) = aabaabab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 25 / 40

  39. Complexity & Sturmian words Christoffel words: Construction by example Lower & Upper Christoffel words of slope 3 5 a a b b a a a b b a a a b b a a L(3,5) = aabaabab U(3,5) = babaabaa Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 26 / 40

  40. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  41. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . √ 5 − 1 Example: y = x − → Fibonacci word 2 b a a b a b a a b a a Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  42. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . √ 5 − 1 Example: y = x − → Fibonacci word 2 b a a b a b a a b a a f = abaababaabaababaaba · · · (note: disregard 1st a in construction) Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  43. Complexity & Sturmian words From Christoffel words to Sturmian words Sturmian words: Obtained *similarly* by replacing the line segment by a half-line: y = α x + ρ with irrational α ∈ ( 0 , 1 ) , ρ ∈ R . √ 5 − 1 Example: y = x − → Fibonacci word 2 b a a b a b a a b a a f = abaababaabaababaaba · · · (note: disregard 1st a in construction) √ 5 − 1 Standard Sturmian word of slope , golden ratio conjugate 2 Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 27 / 40

  44. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  45. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  46. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  47. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Then C w ( n ) is constant on some interval of length s . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  48. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Then C w ( n ) is constant on some interval of length s . Finally C w ( n ) decreases by 1 with each n on an interval of length r . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  49. Trapezoidal words Factor complexity of finite Sturmian words In 1999, A. de Luca studied the behaviour of the factor complexity of finite words. He showed: Theorem (de Luca 1999) If w is a finite Sturmian word of length | w | (i.e., a cyclic shift of a Christoffel word), then the graph of C w ( n ) as a function of n (for 0 ≤ n ≤ | w | ) is that of a regular trapezoid (possibly degenerated to a triangle). That is: C w ( n ) increases by 1 with each n on some interval of length r . Then C w ( n ) is constant on some interval of length s . Finally C w ( n ) decreases by 1 with each n on an interval of length r . So if we set D w ( n ) = C w ( n + 1 ) − C w ( n ) for each n with 0 ≤ n ≤ | w | − 1, then the word D w ( 0 ) D w ( 1 ) · · · D w ( | w | − 1 ) takes the form 1 r 0 s ( − 1 ) r . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 28 / 40

  50. Trapezoidal words Example Graph of the factor complexity of the Christoffel word L ( 3 , 5 ) = aabaabab Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 29 / 40

  51. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  52. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  53. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  54. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. This is because there is 1 factor of length 0, namely the empty word ε . Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  55. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. This is because there is 1 factor of length 0, namely the empty word ε . So any trapezoidal word is on a binary alphabet and the family of trapezoidal words properly contains all finite Sturmian words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  56. Trapezoidal words Trapezoidal words This “trapezoidal property” does not characterise Sturmian words. For example, aabb is trapezoidal, but not Sturmian. Note: If w is a trapezoidal word (i.e., its ‘complexity’ graph has the same behaviour as that of Sturmian words), then necessarily C w ( 1 ) = 2. This is because there is 1 factor of length 0, namely the empty word ε . So any trapezoidal word is on a binary alphabet and the family of trapezoidal words properly contains all finite Sturmian words. F. D’Alessandro (2002): classified all non-Sturmian trapezoidal words. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 30 / 40

  57. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  58. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Theorem (de Luca-G.-Zamboni) Let w be a trapezoidal word. Then w contains | w | + 1 distinct palindromes (including ε ). Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  59. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Theorem (de Luca-G.-Zamboni) Let w be a trapezoidal word. Then w contains | w | + 1 distinct palindromes (including ε ). That is, trapezoidal words (and hence finite Sturmian words) are “rich” in palindromes in the sense that they contain the maximum number of distinct palindromic factors since: Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  60. Trapezoidal words Characterisation of Sturmian palindromes We have shown: Theorem (de Luca-G.-Zamboni) Let w be a palindrome. Then w is Sturmian if and only if w is trapezoidal. Theorem (de Luca-G.-Zamboni) Let w be a trapezoidal word. Then w contains | w | + 1 distinct palindromes (including ε ). That is, trapezoidal words (and hence finite Sturmian words) are “rich” in palindromes in the sense that they contain the maximum number of distinct palindromic factors since: Theorem (Droubay-Justin-Pirillo 2001) A finite word w contains at most | w | + 1 distinct palindromes (including ε ). Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 31 / 40

  61. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  62. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  63. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  64. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  65. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  66. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  67. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  68. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Examples a ω = aaaaaa · · · and ab ω = abbb · · · are rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  69. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Examples a ω = aaaaaa · · · and ab ω = abbb · · · are rich. ( ab ) ω = abababab · · · and ( aba ) ω = abaabaaba · · · are rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  70. Rich words Rich words Definition (G.-Justin 2007) A finite word w is rich iff w contains exactly | w | + 1 distinct palindromes. Examples abac is rich, whereas abca is not rich. The word rich is rich . . . and poor is rich too! Any trapezoidal word is rich, but not conversely. E.g., aabbaa is rich, but not trapezoidal ( C ( 1 ) = 2, C ( 2 ) = 4) Definition (G.-Justin 2007) An infinite word is rich iff all of its factors are rich. Examples a ω = aaaaaa · · · and ab ω = abbb · · · are rich. ( ab ) ω = abababab · · · and ( aba ) ω = abaabaaba · · · are rich. abc is rich, but ( abc ) ω = abcabcabc · · · is not rich. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 32 / 40

  71. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  72. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  73. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  74. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  75. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  76. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  77. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  78. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  79. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

  80. Rich words Rich words . . . Roughly speaking, a finite or infinite word is rich if and only if a new palindrome is introduced at each new position. Example: abaabaaabaaaabaaaaab · · · Amy Glen (Murdoch University) Rich, Sturmian & trapezoidal words September 2010 33 / 40

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