BALANCES AND ABELIAN COMPLEXITY OF A CERTAIN CLASS OF INFINITE TERNARY WORDS
A word u defined over an alphabet A is c-balanced (c is an element of N) if for all pairs of factors v, w of u of the same length and for all letters a. A, the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet A = {L, S, M} and a class of substitutions phi(p) defined by phi(p)(L) = L p S, phi(p)(S) = M, phi(p)(M) = L(p-1)S where p > 1. We prove that the fixed point of phi(p), formally written as phi(infinity)(p)(L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved.
abelian complexity ternary word substitution balance property