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.
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