Shift maps and their variants on inverse limits with set-valued functions (Q1709055)

\textit{W. S. Mahavier} [Topology Appl. 141, No. 1--3, 225--231 (2004; Zbl 1078.54021)] introduced constructions of inverse limits of compact spaces using bonding maps that are upper semi-continuous and set-valued. Both topological and dynamical properties of the resulting spaces have been studied, and this paper continues this with a focus on shift maps and their variants defined on such spaces. A unified approach using the pullback construction is described and used to recover and extend some known results about the limit spaces. In the setting of a single bonding map the bonding map naturally induces a shift map on the limit space, and an example is constructed in which the limit space is homeomorphic to the Cantor set and the dynamical system is a minimal topological subsystem of the full shift on two symbols. Finally topological properties of the limit spaces, and in particular higher homotopical and cohomological connectivity are studied.