Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling σ functions, and lead to exact functional relations for the local flow β functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of σ are sometimes not true fixed points under continuous changes in scale, and zeroes of β do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.

A physical model with a limit cycle added as section IV, along with references