The degree of squares is an atom

We answer an open question in the theory of degrees of infinite sequences with respect to transducibilityby finite-state transducers. An initial study of this partial order of degrees was carried out in [1], but many basic questions remain unanswered.One of the central questions concerns the existence of atom degrees, other than the degree of the ‘identity sequence’ 100101102103⋯ . A degree is called an ‘atom’ if below it there is only the bottom degree 00 , which consists of the ultimately periodic sequences. We show that also the degree of the ‘squares sequence’ 1001011041091016⋯ is an atom.\u3cbr/\u3e\u3cbr/\u3eAs the main tool for this result we characterise the transducts of ‘spiralling’ sequences and their degrees. We use this to show that every transduct of a ‘polynomial sequence’ either is in 00 or can be transduced back to a polynomial sequence for a polynomial of the same order