Instability of Renormalization

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 and unstable manifolds of renormalization are full families of minimal dimension. On the other hand, physically more realistic systems may exhibit renormalization phenomena which are surprisingly different when compared with the classical theory. In phase space one observes the coexistence phenomenon, i.e. even for bounded combinatorial type there are systems whose attractor has bounded geometry but which are topologically conjugate to systems whose attractor has degenerate geometry. In parameter space there is dimensional discrepancy at the renormalization fixed point, i.e. the unstable manifold of the renormalization fixed point contains a strong unstable manifold which is a full family of minimal dimension but the whole unstable manifold has a strictly larger dimension.