A characterisation of the dimension of inverse limits of set-valued functions on intervals
Volume 249 / 2020
Fundamenta Mathematicae 249 (2020), 1-19
MSC: 54F15, 54C60, 54E45, 54H99.
DOI: 10.4064/fm708-5-2019
Published online: 17 October 2019
Abstract
For a sequence $\boldsymbol f$ of upper semicontinuous set-valued functions $f_i:\mathbb I _i\to 2^{\mathbb I _{i-1}}$ (where $\mathbb I _i=[0,1]$ for each $i\in \mathbb N $), we introduce the notion of a weighted sequence. We show that $\dim (\underleftarrow{\lim}\,{\boldsymbol f} )= n$ if and only if the maximal length of a weighted sequence admitted by $\boldsymbol f$ is $2n-2$. Furthermore, $\dim (\underleftarrow{\lim}\,{\boldsymbol f} )\ge n$ if and only if there is an increasing sequence $\langle i_1,\ldots ,i_n\rangle $ such that the projection of $\underleftarrow{\lim}\,{\boldsymbol f} $ to $\prod _{1\le j\le n}\mathbb I _{i_j}$ contains an $n$-cell.
