Pointwise Chain Recurrent Maps Of The Space Y

Let Y {zE C: Z3E [0, 1]}(equipped with subspace topology of the complex plane C) and let f: Y ~Y be a continuous map. We show that if f is pointwise chain recurrent (i.e, every point of Y is chain recurrent under f. ), then one of the following conclusions must hold: (1) f is the identity. (2) f has exactly one fixed point. (3) f2 is the identity, but f is not the identity. (4) f2 is turbulent. The above result is a generalization of a result of Block and Coven for pointwise chain recurrent map of the interval.