  We examine the one-humped map at the period-doubling transition to chaos, and
ask whether its long-term memory is stack-like (last-in, first-out) or
queue-like (first-in, first-out). We show that it can be recognized by a
real-time automaton with one queue, or two stacks, and give several new
grammatical characterizations of it. We argue that its memory has a queue-like
character, since a single stack does not suffice.
  We also show that its dynamical zeta function, generating function and growth
function are transcendental. The same results hold for any period-multiplying
cascade. We suggest that transcendentality might be a sign of dynamical phase
transitions in other systems as well.
