  In this paper we give a new sufficient condition for asymptotic periodicity
of Frobenius-Perron operator corresponding to two--dimensional maps. The result
of the asymptotic periodicity for strictly expanding systems, that is, all
eigenvalues of the system are greater than one, in a high-dimensional dynamical
systems was already known. Our new theorem enables to apply for the system
having an eigenvalue smaller than one. The key idea for the proof is a function
of bounded variation defined by line integration. Finally, we introduce a new
two-dimensional dynamical system exhibiting the asymptotic periodicity with
different periods depending on parameter values, and discuss to apply our
theorem to the model.
