Some properties of the chain recurrent set of dynamical systems

In this paper, the conclusions that the chain-transitive components of a flow on a compact Hausdorff topological space are precisely the components of the chain recurrent set and that each chain-transitive component is chain recurrent are proved. A necessary and sufficient condition for the entire space to be a single chain-transitive component is given at the same time. Moreover some properties about the chain-transitive components are discussed. At last for a continuous map defined on a compact metric space, the strong-epsilon-chain, strong-chain-transitive component are introduced and some existing results are improved. (C) 2002 Elsevier Science Ltd. All rights reserved.