  Timed languages contain sequences of discrete events ("letters'') separated
by real-valued delays, they can be recognized by timed automata, and represent
behaviors of various real-time systems. The notion of bandwidth of a timed
language defined in a previous paper characterizes the amount of information
per time unit, encoded in words of the language observed with some precision
{\epsilon}.
  In this paper, we identify three classes of timed automata according to the
asymptotics of the bandwidth of their languages with respect to this precision
{\epsilon}: automata are either meager, with an O(1) bandwidth, normal, with a
{\Theta}(log (1/{\epsilon})) bandwidth, or obese, with {\Theta}(1/{\epsilon})
bandwidth. We define two structural criteria and prove that they partition
timed automata into these three classes of bandwidth, implying that there are
no intermediate asymptotic classes. The classification problem of a timed
automaton is PSPACE-complete.
  Both criteria are formulated using morphisms from paths of the timed
automaton to some finite monoids extending Puri's orbit graphs; the proofs are
based on Simon's factorization forest theorem.
