The dynamics of continuous maps of finite graphs through inverse limits

Abstract

Suppose that f: G -* G is a continuous piecewise monotone function on a finite graph G. Then the following are equivalent: (i) f has positive topological entropy; (ii) there are disjoint intervals I1 and I2 and a positive integer n with I1 U I2 C fn(JI) n fn(I2); (iii) the inverse limit space constructed by using f on G as a single bonding map contains an indecomposable subcontinuum. This result generalizes known results for the interval and circle.