  In this paper, we consider chaotic dynamics and variational structures of
area-preserving maps. There is a lot of study on the dynamics of their maps and
the works of Poincare and Birkhoff are well-known. To consider variational
structures of area-preserving maps, we define a special class of
area-preserving maps called monotone twist maps. Variational structures
determined from twist maps can be used for constructing characteristic
trajectories of twist maps. Our goal is to prove the existence of an infinite
transition orbit, which represents an oscillating orbit between fixed points
infinite times, through minimizing methods.
