Defects of fixed points of subsitutions

Petr Ambroz

Czech Technical University, Czech Republic

Coauthor(s): Lubomíra Balková and Edita Pelantová

Droubay, Justin and Pirillo showed that every finite word $w$ contains at most $\vert w\vert+1$ palindromes (note that the empty word is also considered as a palindrome), where $\vert w\vert$ denotes the length of $w$. The difference between $\vert w\vert+1$ and the actual number of palindromes $w$ contains is called defect of $w$. A finite word containing the maximal possible number of palindromes, i.e., having zero defect, is called full. An infinite word is said to be full if all its prefixes are full.

We prove fullness of infinite words coding the distances between neighboring $\beta$-integers, both in the simple and in the non-simple Parry case. We show that similar technique can also be used to prove fullness of some other words being fixed points of substitutions, e.g. the Rote substitution or the period-doubling substitution.