Lie complexity of words

Abstract

Given a finite alphabet Σ and a right-infinite word w over Σ, we define the Lie complexity function Lw : N → N, whose value at n is the number of conjugacy classes (under cyclic shift) of length-n factors x of w with the property that every element of the conjugacy class appears in w.We show that the Lie complexity function is uniformly bounded for words with linear factor complexity, and as a result we show that words of linear factor complexity have at most finitely many primitive factors y with the property that y n is again a factor for every n.We then look at automatic sequences and show that the Lie complexity function of a k-automatic sequence is again k-automatic.

Topological invariants for words of linear factor complexity

“…In fact, finiteness of Per(w) holds for every word of linear factor complexity [3], without requiring the assumption that w be recurrent, although the bound will not, in general, be as good as the one we obtain in the recurrent case.…”

“…Upper bounds for Per(w) have been obtained previously [3], where it is shown that #Per(w) ≤ lim sup n→∞ p w (n + 1) − p w (n) + 1, which a result of Cassaigne [7] shows is finite when w has linear factor complexity. Bounds for Per(w) had been previously considered in other contexts [11].…”

“…An important feature in the study of combinatorics of words is the search for meaningful invariants, which give insight into the underlying complexity and structure of given words. There are numerous examples of such invariants in the theory, such as the critical exponent [5], cyclic complexity [8], arithmetical complexity [2], abelian complexity [10], Lie complexity [3], Lempel-Ziv complexity [12], letter and word frequencies [1,Chapt. 1], and the factor complexity [1,Chapt.…”

Automatic Sequences of Rank Two

“…The following result appears in earlier work of the authors [3, Theorem 1.2], but we give a simpler and more self-contained proof here. We note, however, that the following proof does not recover the upper bound on the number of primitive factors that can occur with unbounded exponent (up to cyclic equivalence), which was given in [3].…”

“…Kobayashi and Otto [18] gave an efficient algorithm to test the repetitivity of a pure morphic word. Klouda and Starosta [15] showed further that every pure morphic word x is discrete, while the authors [3] proved that words of linear factor complexity are discrete.…”