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Title:
Renormalization approach to neutral dynamics
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Abstract:
For a complex one-dimensional dynamical system, a periodic point is called neutral if it is neither attracting nor repelling. There are always some orbits that ``rotate'' around a neutral periodic point. However, the topology and exact dynamics of the set of orbits that ``rotate'' are delicate questions. The Siegel case, when all close orbits rotate along circles, is related to the classical small divisor problem in the KAM theory. We will give an overview of known results and conjectures on neutral dynamics. In particular, we will discuss the conjectural structure of the maximal ``rotation set'' around a neutral periodic point as well as semicontinuous stability of such set under a small perturbation. The conjectures are verified in certain cases, thanks to different renormalization theories.
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