A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
A square root map on Sturmian words

Julkaisun tekijät: Jarkko Peltomäki,Markus A. Whiteland
Kustantaja: Electronic Journal of Combinatorics
Julkaisuvuosi: 2017
Journal: The Electronic Journal of Combinatorics
Volyymi: 24
Julkaisunumero: 1
eISSN: 1077-8926

Tiivistelmä

We introduce a square root map on Sturmian words and study its
properties. Given a Sturmian word of slope \$alpha\$, there exists
exactly six minimal squares in its language (a minimal square does not
have a square as a proper prefix). A Sturmian word \$s\$ of slope \$alpha\$
can be written as a product of these six minimal squares: \$s = X_1^2
X_2^2 X_3^2 cdots\$. The square root of \$s\$ is defined to be the word
\$sqrt{s} = X_1 X_2 X_3 cdots\$. The main result of this paper is that
\$sqrt{s}\$ is also a Sturmian word of slope \$alpha\$. Further,
we characterize the Sturmian fixed points of the square root map, and we
describe how to find the intercept of \$sqrt{s}\$ and an occurrence of
any prefix of \$sqrt{s}\$ in \$s\$. Related to the square root map, we characterize the solutions of the word equation \$X_1^2 X_2^2 cdots
X_n^2 = (X_1 X_2 cdots X_n)^2\$ in the language of Sturmian words of
slope \$alpha\$ where the words \$X_i^2\$ are minimal squares of slope
\$alpha\$.

We also study the square root map in a more general
setting. We explicitly construct an infinite set of non-Sturmian fixed
points of the square root map. We show that the subshifts \$Omega\$
generated by these words have a curious property: for all \$w in Omega\$
either \$sqrt{w} in Omega\$ or \$sqrt{w}\$ is periodic. In
particular, the square root map can map an aperiodic word to a periodic
word.

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Last updated on 2019-12-06 at 16:14