Shrinking without doing much at all

In 1952 Bing astonished the mathematical world with his wild involution on S 3 .It has been among the most seminal examples in topology.The example depends on finding shrinking homeomorphisms of Bing's decomposition of S 3 into points and arcs.If Bing's original homeomorphisms are varied, Bing's original wild involution changes by conjugation, which preserves some analytic properties while altering others.In 1988, Bing published a second paper, Shrinking without lengthening, answering a question that one of the present authors posed to him in an effort to understand the geometry of the entire conjugacy class.Here we produce a counterintuitive construction, namely a method to shrink the Bing decomposition doing almost nothing at all: neither lengthening much nor rotating much.