Abstract
Phys.Rev.D83:065019,2011 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 {\sigma} functions, and
lead to exact functional relations for the local flow {\beta} functions, whose
solutions may have novel, exotic features, including multiple branches. As a
result, fixed points of {\sigma} are sometimes not true fixed points under
continuous changes in scale, and zeroes of {\beta} do not necessarily signal
fixed points of the flow, but instead may only indicate turning points of the
trajectories.
