Shrinking of toroidal decomposition spaces

Given a sequence of oriented links L1,L2,L3,… each of which has a distinguished, unknotted component, there is a decomposition space D of S3 naturally associated to it, which is constructed as the components of the intersection of an infinite sequence of nested solid tori. The Bing and Whitehead continua are simple, well known examples. We give a necessary and sufficient criterion to determine whether D is shrinkable, generalising previous work of F. Ancel and M. Starbird and others. This criterion can effectively determine, in many cases, whether the quotient map S3→S3/D can be approximated by homeomorphisms