## Institute for Mathematical Sciences

## Preprint ims17-01

** Marco Martens and Bjorn Winkler**
*Instability of renormalization*

Abstract: 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.

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