Linearization and circle diffeomorphisms

Today, i have learn that there is quite interesing interplay between the theory of linearization of maps f(z)=\lambda z + O(z^2) and analytic circle diffeomorphism theory.

In fact, according to results of P.Marco, for every such f(z) there is a fully invariant set K, which is called a Siegel continua, or Hedgehog (in the Cremer case) such that if we consider the conformal representation of the complement \bar{C} - K to the complement of the unit disk, then it conjugates the map f(z) to an anylitc circle diffeomorphism, which actually even has the same roattion number.And the last property is crucial!

The idea of the proof of the existence of such K is basically the following: in case of the rational multiplier \lambda there are quite a lot of things known about the topology in the neighborhood of the fixed point 0:  Leau-Fatou flower theorem. Which states that one can construct attracting and repelling petals, all have 0 on the boundary, which interchange and form the whole neighborhood of 0.  As a candidate for K in the rational case one can take the union of intersections of attracting and repelling petals: they are clearly fully invariant and the other properties can also be established. Then it left to use the density argument and show that the density of rational translates to our case in terms of existence of these continua K.

It follows that we have a dictionary between these two theories.  Even though a lot of things are the same in this dictionary, there are differences. For example (the only one i know so far), the set of lineariziable quadratic maps with irrationally indifferent fixed point – is the set of Bruno numbers (one way – Bruno, Siegel,  the necessity – Yoccoz). However, for circle diffeomorphisms, the linearizability condition is for the rotation number to be Herman!

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