tailieunhanh - Báo cáo toán học: "Critical exponents of words over 3 letters"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Critical exponents of words over 3 letters. | Critical exponents of words over 3 letters Elise Vaslet Institut de Mathematiques de Luminy Université Aix-Marseille II Marseille France vaslet@ Submitted Oct 5 2010 Accepted May 25 2011 Published Jun 6 2011 Mathematics Subject Classification 68R15 Abstract For all a RT 3 where RT 3 7 4 is the repetition threshold for the 3-letter alphabet there exists an infinite word over 3 letters whose critical exponent is a. 1 Introduction Let A be a finite alphabet. Any finite word v over A v e can be factorized as v pke where - k 1 - e is a prefix of p - p is minimal We then say that v has period p excess e and exponent E v v p . For example the English word church has period chur excess ch and exponent 3 2 while the French word entente has period ent excess e and exponent 7 3. A finite or infinite word is said to be a-free resp. a -free if none of its subwords has exponent ft with ft a resp. ft a . The critical exponent of an infinite word w over A is defined as Ec w sup E v G Q v subword of w . For example the binary word abab2 abn has critical exponent to. The Thue-Morse word fixed point of the morphism 0 01 1 10 has critical exponent 2 10 and 1 . The Fibonacci word fixed point of the morphism 0 01 1 0 has critical exponent 2 Ộ where Ộ is the golden number 8 . The problem to determine if for a given real number a 1 there is an infinite word wa with critical exponent a has been solved by Krieger and Shallit 7 . The number of THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P125 1 letters they use to construct the word wa grows very fast as a tends to 1 and they left open the construction of wa over an alphabet with a fixed size. Let k be a natural number and let Ak be the k-letter alphabet. The repetition threshold on k letters is the real number see 6 and 3 for more details RT k inf Ec w w G Ak . Dejean 6 conjectured that Í 2 if k 2 RT k 7 5 if k 3 if k 4 k k - 1 if k 4 and this conjecture is now proved see 9 and 5 . It is clear that if a RT k no word over .

• Báo cáo khoa học
• báo cáo của tạp chí Department of Mathematic
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học