  For each piecewise monotonic map tau of [0,1], we associate a pair of
C*-algebras F_tau and O_tau and calculate their K-groups. The algebra F_tau is
an AI-algebra. We characterize when F_tau and O_\tau are simple. In those
cases, F_tau has a unique trace, and O_tau is purely infinite with a unique
KMS-state. In the case that tau is Markov, these algebras include the
Cuntz-Krieger algebras O_A, and the associated AF-algebras F_A. Other examples
for which the K-groups are computed include tent maps, quadratic maps,
multimodal maps, interval exchange maps, and beta-transformations. For the case
of interval exchange maps and of beta-transformations, the C*-algebra O_tau
coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani
respectively.
