36 research outputs found
Cyclic Complexity of Words
We introduce and study a complexity function on words called
\emph{cyclic complexity}, which counts the number of conjugacy classes of
factors of length of an infinite word We extend the well-known
Morse-Hedlund theorem to the setting of cyclic complexity by showing that a
word is ultimately periodic if and only if it has bounded cyclic complexity.
Unlike most complexity functions, cyclic complexity distinguishes between
Sturmian words of different slopes. We prove that if is a Sturmian word and
is a word having the same cyclic complexity of then up to renaming
letters, and have the same set of factors. In particular, is also
Sturmian of slope equal to that of Since for some
implies is periodic, it is natural to consider the quantity
We show that if is a Sturmian word,
then We prove however that this is
not a characterization of Sturmian words by exhibiting a restricted class of
Toeplitz words, including the period-doubling word, which also verify this same
condition on the limit infimum. In contrast we show that, for the Thue-Morse
word , Comment: To appear in Journal of Combinatorial Theory, Series
Open and closed complexity of infinite words
In this paper we study the asymptotic behaviour of two relatively new
complexity functions defined on infinite words and their relationship to
periodicity. Given a factor of an infinite word with
each belonging to a fixed finite set we say is closed
if either or if is a complete first return to some factor
of Otherwise is said to be open. We show that for an aperiodic
word the complexity functions (resp.
that count the number of closed (resp. open) factors of of each
given length are both unbounded. More precisely, we show that if is
aperiodic then and for any syndetic subset of However,
there exist aperiodic infinite words verifying
Keywords: word complexity, periodicity, return words
On Infinite Prefix Normal Words
Prefix normal words are binary words that have no factor with more s than
the prefix of the same length. Finite prefix normal words were introduced in
[Fici and Lipt\'ak, DLT 2011]. In this paper, we study infinite prefix normal
words and explore their relationship to some known classes of infinite binary
words. In particular, we establish a connection between prefix normal words and
Sturmian words, between prefix normal words and abelian complexity, and between
prefix normality and lexicographic order.Comment: 20 pages, 4 figures, accepted at SOFSEM 2019 (45th International
Conference on Current Trends in Theory and Practice of Computer Science,
Nov\'y Smokovec, Slovakia, January 27-30, 2019