ON NON-PERIODIC SOLUTIONS OF INDEPENDENT SYSTEMS OF WORD EQUATIONS OVER THREE UNKNOWNS

摘要

We investigate the open question asking whether there exist independent systems of three equations over three unknowns admitting non-periodic solutions, formulated in 1983 by Culik II and Karhumaki. In particular, we give a negative answer to this question for a large class of systems. More specifically, the question remains open only for a well specified class of systems. We also investigate systems of two equations over three unknowns for which we give necessary and sufficient conditions for admitting at most quasi-periodic solutions, i.e., solutions where the images of two unknowns are powers of a common word. In doing so, we also give a number of examples showing that these conditions represent a boundary point between systems admitting purely non-periodic solutions and those admitting at most quasi-periodic ones.