  A new singular perturbation method based on the Lie symmetry group is
presented to a system of difference equations. This method yields consistent
derivation of a renormalization group equation which gives an asymptotic
solution of the difference equation. The renormalization group equation is a
Lie differential equation of a Lie group which leaves the system approximately
invariant. For a 2-D symplectic map, the renormalization group equation becomes
a Hamiltonian system and a long-time behaviour of the symplectic map is
described by the Hamiltonian. We study the Poincar\'e-Birkoff bifurcation in
the 2-D symplectic map by means of the Hamiltonian and give a condition for the
bifurcation.
