The inverse limit space and the dynamics of a graph map

Let f :G→G be a strictly piecewise monotone continuous map on a finite graph G. By investigating the topological structure of the inverse limit space (G,f) using f as a sole bonding map, we show that the following statements are equivalent: ,f) contains no indecomposable subcontinuum. e topological entropy of f is zero. ,f) is Suslinean. ch homeomorphism of (G,f) has zero topological entropy. has finitely many nontrivial minimal sets. e set of recurrent points of f is closed. ch ω-limit point is a recurrent one. ch recurrent point of f is an almost periodic one.