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

Abstract

Inverse limit spaces of compacta with upper semi-continuous compact set-valued functions are studied via shift maps and their variants. We give a representation of such spaces as the limits of ordinary inverse sequences, which allows us to prove some known results and their extensions in a unified scheme. Next we present a scheme to construct shift dynamics on the inverse limit space with various dynamical features. In particular we construct an inverse sequence over [0,1] with a single upper semi-continuous function f as its bonding function such that (i) the inverse limit space [0,1]f is homeomorphic to the Cantor set and (ii) the shift map σf:[0,1]f→[0,1]f is topologically conjugate to a minimal subshift of a Bernoulli full shift. Also we study local/global connectivity of the inverse limit space over a compactum with a single upper semi-continuous bonding function in terms of homotopy/(co)homology groups, again via shift maps and their variants.