  We study finite-state transducers and their power for transforming infinite
words. Infinite sequences of symbols are of paramount importance in a wide
range of fields, from formal languages to pure mathematics and physics. While
finite automata for recognising and transforming languages are well-understood,
very little is known about the power of automata to transform infinite words.
  The word transformation realised by finite-state transducers gives rise to a
complexity comparison of words and thereby induces equivalence classes, called
(transducer) degrees, and a partial order on these degrees. The ensuing
hierarchy of degrees is analogous to the recursion-theoretic degrees of
unsolvability, also known as Turing degrees, where the transformational devices
are Turing machines. However, as a complexity measure, Turing machines are too
strong: they trivialise the classification problem by identifying all
computable words. Finite-state transducers give rise to a much more
fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly
anything is known about transducer degrees, in spite of their naturality.
  We use methods from linear algebra and analysis to show that there are
infinitely many atoms in the transducer degrees, that is, minimal non-trivial
degrees.
