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

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.