  We explore fundamental questions about the renormalization group through a
detailed re-examination of Feigenbaum's period doubling route to chaos. In the
space of one-humped maps, the renormalization group characterizes the behavior
near any critical point by the behavior near the fixed point. We show that this
fixed point is far from unique, and characterize a submanifold of fixed points
of alternative RG transformations. We build on this framework to systematically
distinguish and analyze the allowed singular and `gauge' (analytic and
redundant) corrections to scaling, explaining numerical results from the
literature. Our analysis inspires several conjectures for critical phenomena in
statistical mechanics.
