  Numerical characteristics of polynomial identities of left nilpotent algebras
are examined. Previously, we came up with a construction which, given an
infinite binary word, allowed us to build a two-step left nilpotent algebra
with specified properties of the codimension sequence. However, the class of
the infinite words used was confined to periodic words and Sturm words. Here
the previously proposed approach is generalized to a considerably more general
case. It is proved that for any algebra constructed given a binary word with
subexponential function of combinatorial complexity, there exists a
PI-exponent, and its precise value is computed.
