On dimension and shape of inverse limits with set-valued functions (Q2836011)

Consider compacta \(X_i\) and upper semicontinuous maps \(f_i : X_i \to 2^{X_{i+1}}\) for all \(i\in \mathbb{N}\). This paper studies the dimension and shape of the inverse limit \(X=\displaystyle \lim_{\longleftarrow} (X_i, f_i)\). In order to determine good bounds on the dimension of \(X\), the author introduces a very nice idea of expand-contract sequences. These are, roughly speaking, finite sequences of the following kind:NEWLINENEWLINE(1) start with a point;NEWLINENEWLINE(2) the image of this point via (possibly numerous consecutive) \(f_i\)'s should be a subspace of dimension at least \(1\);NEWLINENEWLINE(3) the image of the subspace of (2) via (possibly numerous consecutive) \(f_i\)'s should be a point;NEWLINENEWLINE(4) repeat the pattern, ending with a single point as in (3).NEWLINENEWLINEThe maximal cumulative change of dimensions generated by such a sequence is denoted by \(\widetilde J\) and the main theorem states that \(\dim(X) \leq \widetilde J + \sup_i \{\dim(X_i)\}\), which is a significant improvement of previous results on the dimension of inverse limits. In certain \(1\)-dimensional cases the author also proves the bound \(\dim(X) \geq \widetilde J\), which determines the dimension almost completely and may be applied to many examples studied in the literature. The paper is concluded by some results on the shape of \(X\).