  In the theory of renormalization for classical dynamical systems, e.g.
unimodal maps and critical circle maps, topological conjugacy classes are
stable manifolds of renormalization. Physically more realistic systems on the
other hand may exhibit instability of renormalization within a topological
class. This instability gives rise to new phenomena and opens up directions of
inquiry that go beyond the classical theory. In phase space it leads to the
coexistence phenomenon, i.e. there are systems whose attractor has bounded
geometry but which are topologically conjugate to systems whose attractor has
degenerate geometry; in parameter space it causes dimensional discrepancy, i.e.
a topologically full family has too few dimensions to realize all possible
geometric behavior.
