Logarithmic Equal-Letter Runs for BWT of Purely Morphic Words

String Attractors and Infinite Words

“…In [10] the number r of BWT equal-letter runs has been studied for all prefixes obtained by iterating a morphism. In [7] the measure z x (n) that gives the number z of phrases in the LZ77 parsing of the prefix x[1, n] has been studied.…”

“…Suppose now x is aperiodic. For morphisms defined on a binary alphabet, it holds that if x = µ ∞ (a) is aperiodic, then |µ i (a)| grows exponentially with respect to i (see [10]). Moreover, if µ is primitive, then by [8, Theorem 1] and [1, Theorem 10.9.4] x is linearly recurrent, and by Theorem 2 we have that s x (n) = Θ(1).…”

“…Moreover, if µ is primitive, then by [8, Theorem 1] and [1, Theorem 10.9.4] x is linearly recurrent, and by Theorem 2 we have that s x (n) = Θ(1). If µ is not primitive, as summed up in [10], then only one of the following cases occurs: (1) there exist a coding τ : Σ → {a, b} + and a primitive morphism ϕ : [25]; (2) x contains arbitrarly large factors on {b} * . For case (1), since τ preserves the recurrence of a word and that ϕ ∞ (a) is linearly recurrent, then x is linearly recurrent as well, and by Theorem 2 s x (n) = Θ(1).…”

“…we can further deduce an upper bound for the string attractor profile function and it follows that s x (n) = Θ(log n). Finally, from a classification in [10] we can decide, only from µ, if either s x (n) = Θ(1) or s x (n) = Θ(log n).…”