Equations on palindromes and circular words

In this paper we consider several types of equations on words, motivated by the attempt of characterizing the class of polyominoes that tile the plane by translation in two distinct ways. Words coding the boundary of these polyominoes satisfy an equation whose solutions are in bijection with a subset of the solutions of equations of the form ABA@?B@?=XYX@?Y@?. It turns out that the solutions are strongly related to local periodicity involving palindromes and conjugate words.

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