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

✍ Scribed by Xiangdong Ye

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 986 KB
• Volume: 91
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00188-0

✦ Synopsis

Let I be a closed interval and f : I -I be continuous. We investigate the structure of the inverse limit space lim{l. j'} which contains no indecomposable subcontinuum. In particular, we show that the set of nondegenerate maximal nowhere dense subcontinua of lim{l, 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:

(I) lim{l, f} contains no indecomposable subcontinuum.

(2) The topological entropy of f is zero.

(3) lim{l, f} is Suslinean. (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.