The authors investigate the topological properties of inverse limit spaces where every one-step bonding map \(f\) is the same unimodal map of the interval \({0,1},\) i.e. a map with only one critical point.
Sufficient combinatorial conditions on the map are given for a subcontinuum of the inverse limit space to be either an arc or an arc plus a limiting ray.
The following is also proven.
Theorem: Let \(\mathcal{F}\) be a finite or countable collection of unimodal maps, each having a periodic critical point. There exists a map \(g\) such that for each \(f \in \mathcal{F}\) the inverse limit space \((I,g)\) contains a subcontinuum \(H(f)\) homeomorphic to \((I,f).\) Every proper subcontinuum \(H\) of \((I,g)\) which is not an arc or an arc plus limiting ray is homeomorphic to \((I,f)\) or \(([c_2,c_1],f)\) for some \(f \in \mathcal{F}.\)
Recently Kailhofer [A partical classification of inverse limit spaces of tent maps with periodic critical points, preprint] and Raines [Inverse limits of Markov maps of simple graphs, preprint] have independently obtained further results on the topological structure of inverse limits of \([0,1]\) with unimodal bonding maps and methods of distinguishing the inverse limit spaces resulting from various maps.