  We propose new methods to extend the renormalization group transformation to
complex coupling spaces. We argue that the Fisher's zeros are located at the
boundary of the complex basin of attraction of infra-red fixed points. We
support this picture with numerical calculations at finite volume for
two-dimensional O(N) models in the large-N limit and the hierarchical Ising
model. We present numerical evidence that, as the volume increases, the
Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a
Wilson action, stabilize at a distance larger than 0.15 from the real axis in
the complex beta=4/g^2 plane. We discuss the implications for proofs of
confinement and searches for nontrivial infra-red fixed points in models beyond
the standard model.
