  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.
