On winning shifts of marked uniform 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 α1, α2, . . . of integers such that on round n the player A chooses a subset Sn of size αn of some fixed finite alphabet and the player B picks a letter from the set Sn. 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. It is known that W(X) and X have the same factor complexity. 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.