Compacta admitting retractions close to the identity map

Abstract

We develop techniques for determining certain structural properties of inverse limits on compacta. In particular, we show if easily observable properties of the bonding mappings are present, then one can identify nested sequences of subcompacta of the inverse limit space whose members are copies of subcompacta of the factor spaces, and a sequence of retractions of the inverse limit space onto members of this nested sequence that converges uniformly to the identity mapping. We discuss applications to inverse limits on continua, and in the special case of a continuum $X$ that admits such a sequence of retractions onto arcs, we establish properties that describe the nature of proper subcontinua of $X$.