  In this paper, we first show that any nonlinear monotonic increasing
contracting maps with one discontinuous point on a unit interval which has an
unique periodic point with period $n$ conjugates to a piecewise linear
contracting map which has periodic point with same period. Second, we consider
one parameter family of monotonic increasing contracting maps, and show that
the family has the periodic structure called Arnold tongue for the parameter
which is associated with the Farey series. This implies that there exist a
parameter set with a positive Lebesgue measure such that the map has a periodic
point with an arbitrary period. Moreover, the parameter set with period $(m+n)$
exists between the parameter set with period $m$ and $n$.
