A1 Refereed original research article in a scientific journal
On winning shifts of marked uniform substitutions
Authors: Peltomäki Jarkko, Salo Ville
Publisher: E D P Sciences
Publication year: 2019
Journal: RAIRO: Informatique Théorique et Applications / RAIRO: Theoretical Informatics and Applications
Volume: 53
Issue: 1-2
First page : 51
Last page: 66
Number of pages: 16
ISSN: 0988-3754
eISSN: 1290-385X
DOI: https://doi.org/10.1051/ita/2018007
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/39569288
Abstract
The second author introduced with I. Törmä a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has a predetermined (possibly finite) choice sequence $\alpha_1$, $\alpha_2$, $\ldots$ of integers such that on round $n$ the player $A$ chooses a subset $S_n$ of size $\alpha_n$ of some fixed finite alphabet and the player $B$ picks a letter from the set $S_n$. The outcome is determined by whether the word obtained by concatenating the letters $B$ picked lies in a prescribed target set $X$ (a win for player $A$) or not (a win for player $B$). Typically, we consider $X$ to be a subshift. The winning shift $W(X)$ of a subshift $X$ is defined as the set of choice sequences for which $A$ has a winning strategy when the target set is the language of $X$. The winning shift $W(X)$ mirrors some properties of $X$. For instance, $W(X)$ and $X$ have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that $W(X)$ and $X$ have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.
The second author introduced with I. Törmä a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has a predetermined (possibly finite) choice sequence $\alpha_1$, $\alpha_2$, $\ldots$ of integers such that on round $n$ the player $A$ chooses a subset $S_n$ of size $\alpha_n$ of some fixed finite alphabet and the player $B$ picks a letter from the set $S_n$. The outcome is determined by whether the word obtained by concatenating the letters $B$ picked lies in a prescribed target set $X$ (a win for player $A$) or not (a win for player $B$). Typically, we consider $X$ to be a subshift. The winning shift $W(X)$ of a subshift $X$ is defined as the set of choice sequences for which $A$ has a winning strategy when the target set is the language of $X$. The winning shift $W(X)$ mirrors some properties of $X$. For instance, $W(X)$ and $X$ have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that $W(X)$ and $X$ have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.
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