  The field theoretical renormalization group equations have many common
features with the equations of dynamical systems. In particular, the manner how
Callan-Symanzik equation ensures the independence of a theory from its
subtraction point is reminiscent of self-similarity in autonomous flows towards
attractors. Motivated by such analogies we propose that besides isolated fixed
points, the couplings in a renormalizable field theory may also flow towards
more general, even fractal attractors. This could lead to Big Mess scenarios in
applications to multiphase systems, from spin-glasses and neural networks to
fundamental string (M?) theory. We consider various general aspects of such
chaotic flows. We argue that they pose no obvious contradictions with the known
properties of effective actions, the existence of dissipative Lyapunov
functions, and even the strong version of the c-theorem. We also explain the
difficulties encountered when constructing effective actions with chaotic
renormalization group flows and observe that they have many common virtues with
realistic field theory effective actions. We conclude that if chaotic
renormalization group flows are to be excluded, conceptually novel no-go
theorems must be developed.
