On inverse limit spaces of maps of an interval with zero entropy

Let I be a closed interval and f : I → I be continuous. We investigate the structure of the inverse limit space lim{I, f} which contains no indecomposable subcontinuum. In particular, we show that the set of nondegenerate maximal nowhere dense subcontinua of lim{I, f} is finite if f is piecewise monotone with zero topological entropy. Applying the above result, we show that if f : I → I is piecewise monotone, then the following statements are equivalent: 1. m{I, f} contains no indecomposable subcontinuum. e topological entropy of f is zero. m{I, f} is Suslinean. ch homeomorphism of lim{I, f} has zero topological entropy. o show how the order of lim{I, f} is dependent on the set of periods of f when f is piecewise monotone with zero topological entropy.