The structure of the set of cube-free $Z$-words in a two-letter alphabet

The object of our study is the set of $Z$-words, that is, (bi)infinite sequences of alphabetic symbols indexed by integers. We consider an ordered family of subsets of the set of all the cube-free $Z$-words in a two-letter alphabet. The construction of this family is based on the notion of the local exponent of a $Z$-word. The problem of existence of cube-free $Z$-words which are $Z$-words of local exponent 2 (the minimum possible) is described. An important distinction is drawn between strongly cube-free $Z$-words and $Z$-words of greater local exponent.