  In this paper, several fundamental facts, especially the existence and
uniqueness of an absolutely continuous ergodic measure with an exponential
mixing rate, are derived for smooth expanding circle maps. Although the results
are classical, the coupling method is relatively modern and employed here for
the first time to yield explicit bounds in terms of system constants. The work
constitutes a part of the author's Bachelor's thesis. The author hopes that the
manuscript can serve as a useful introduction to the flexible coupling method
in the theory of dynamical systems.
