Generalized inverse limits and the intermediate value property.

We introduce and discuss various notions of the intermediate value property applicable to upper-semicontinuous set-valued functions f : [0, 1] → 2^[0,1]. In the first part, we present sufficient conditions such that an inverse limit of a sequence of bonding functions of this type is a continuum. In the second part, we examine the relationship between the dynamics of an upper-semicontinuous function with the intermediate value property and the topological structure of the corresponding inverse limit. In particular, we present conditions under which the existence of a cycle of period not a power of 2 implies indecomposability in the inverse limit and vice-versa. Lastly, we show that these conditions are sharp by constructing a family of upper-semicontinuous functions with the intermediate value property and cycles of all periods, yet admits a hereditarily decomposable inverse limit.