A Hereditarily Decomposable Generalized Inverse Limit from a Function on [0,1] with cycles of all periods

TL;DR

This paper constructs a hereditarily decomposable inverse limit from a function on [0,1] with cycles of all periods, challenging previous assumptions about the relationship between periodic points and indecomposability.

Contribution

It demonstrates that an upper semicontinuous, surjective map with all periods can produce a hereditarily decomposable inverse limit, refining conditions for indecomposability.

Findings

Constructed a map with all periods but hereditarily decomposable inverse limit

Showed the sharpness of the 'almost nonfissile' condition

Expanded understanding of periodic points and inverse limit structures

Abstract

In this paper, we consider inverse limits of $[0,1]$ using upper semicontinuous set-valued functions. We aim to expand on a previous paper exploring the relationship between the existence periodic points of a continuous function to the existence of indecomposable subcontinua of the corresponding inverse limit. In a previous paper, sufficient conditions were given such that if a satisfactory bonding map $F$ had a periodic cycle of period not a power of 2, then $\lim\limits_{\leftarrow}\{[0,1],F\}$ contains an indecomposable continuum. We show that the condition that $F$ is almost nonfissile is sharp by constructing an upper semicontinuous, surjective map $F$ that has the intermediate value property and periodic cycles of every period, yet produces a hereditarily decomposable inverse limit.