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

Abstract

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. (1) lim{ I, f} contains no indecomposable subcontinuum. 2. (2) The topological entropy of f is zero. 3. (3) lim{ I, f} is Suslinean. 4. (4) Each homeomorphism of lim{ I, f} has zero topological entropy. We also 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.