  Geometric models and Teichm\"uller structures have been introduced for the
space of smooth expanding circle endomorphisms and for the space of uniformly
symmetric circle endomorphisms. The latter one is the completion of the
previous one under the Techm\"uller metric. Moreover, the spaces of geometric
models as well as the Teichm\"uller spaces can be described as the space of
H\"older continuous scaling functions and the space of continuous scaling
functions on the dual symbolic space. The characterizations of these scaling
functions have been also investigated. The Gibbs measure theory and the dual
Gibbs measure theory for smooth expanding circle dynamics have been viewed from
the geometric point of view. However, for uniformly symmetric circle dynamics,
an appropriate Gibbs measure theory is unavailable, but a dual Gibbs type
measure theory has been developed for the uniformly symmetric case. This
development extends the dual Gibbs measure theory for the smooth case from the
geometric point of view. In this survey article, We give a review of these
developments which combines ideas and techniques from dynamical systems,
quasiconformal mapping theory, and Teichm\"uller theory. There is a
measure-theoretical version which is called $g$-measure theory and which
corresponds to the dual geometric Gibbs type measure theory. We briefly review
it too.
