Combinatorical and Algebraic Structures Seminar
Session details
| Date: | 21.4.2006 |
| Speaker: | Petr Ambrož, FNSPE, Czech Technical University |
| Title: | Palindromic Complexity of Infinite Words Associated with Simple Parry Numbers |
| Abstract: | (spoluautoři: C. Frougny, Z. Masáková and E. Pelantová) A simple Parry number is a real number $\beta > 1$ such that the Renyi expansion of $1$ is finite, of the form $d_{\beta}(1) = t_1 \cdots t_m$. We study the palindromic structure of infinite aperiodic words $u_{\beta}$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word $u_{\beta}$ contains infinitely many palindromes if and only if $t_1 = t_2 = \cdots = t_{m-1} \geq t_m$. Numbers $\beta$ satisfying this condition are the so-called confluent Pisot numbers. We show that if $\beta$ is a confluent Pisot number then $P(n+1) + P(n) = C(n+1) - C(n) + 2$, where $P(n)$ is the palindromic complexity and $C(n)$ is the subword complexity. We then give a complete description of the set of palindromes, its structure and properties. |
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