Relation between Levinson center, chain recurrent set and center of Birkhoff for compact dissipative dynamical systems

In this paper we prove the analogues of Birkhoff's theorem for one-sided dynamical systems (both with continuous and discrete times) with noncompact space having a compact global attractor. The relation between Levinson center, chain recurrent set and center of Birkhoff is established for compact dissipative dynamical systems.