THE DYNAMICS OF CONTINUOUS-MAPS OF FINITE GRAPHS THROUGH INVERSE LIMITS

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 or I2 subset-or-equal-to f(n)(I1) and f(n)(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.