- 06-89 Hans Koch
- Renormalization of Vector Fields
Mar 22, 06
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Abstract. These notes cover some of the recent developments
in the renormalization of quasiperiodic flows.
This includes skew flows over tori, Hamiltonian flows,
and other flows on $\torus^d\times\real^\ell$.
After stating some of the problems and
describing alternative approaches, we focus on the definition
and basic properties of a single renormalization step.
A second part deals with the construction of conjugacies and
invariant tori, including shearless tori, and non-differentiable
tori for critical Hamiltonians. Then we discuss properties related
to the spectrum of the linearized renormalization transformation,
such as the accumulation rates for sequences of closed orbits.
The last part describes extensions from "self-similar" to Diophantine
rotation vectors. This involves sequences of renormalization
transformations that are related to continued fractions
expansions in one and more dimensions.
Whenever appropriate, the discussion of details is restricted
to special cases where inessential technical complications
can be avoided.