Tree maps having chain movable fixed points

In this paper we discuss some basic properties of chain reachable sets and chain equivalent sets of continuous maps. It is proved that if f : T → T is a tree map which has a chain movable fixed point v, and the chain equivalent set CE ( v , f ) is not contained in the set P ( f ) of periodic points of f, then there exists a positive integer p not greater than the number of points in the set End ( [ CE ( v , f ) ] ) − P v ( f ) such that f p is turbulent, and the topological entropy h ( f ) ⩾ ( log 2 ) / p . This result generalizes the corresponding results given in Block and Coven (1986) [2], Guo et al. (2003) [6], Sun and Liu (2003) [10], Ye (2000) [11], Zhang and Zeng (2004) [12]. In addition, in this paper we also consider metric spaces which may not be trees but have open subsets U such that the closures U ¯ are trees. Maps of such metric spaces which have chain movable fixed points are discussed.