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Linéarisation des perturbations holomorphes des rotations et applications. (Linearization of holomorphic perturbations of rotations and applications). (French) Zbl 0929.37017

Let us recall the problem about local linearization of analytic circle diffeomorphisms: for a given rotation number, decide whether a diffeomorphism with this rotation number and sufficiently close to the linear rotation is analytically conjugated to the rotation. Works of Arnold, Rüssmann and Herman provided the affirmative answer when the rotation number is diophantine. The present paper shows that the Brjuno condition is sufficient, and indeed necessary in view of counterexamples of Yoccoz.
In fact, the setting of the paper is more general. One considers a holomorphic map \(F\) defined on an annulus and close to the rotation by \(\alpha\). Then there is a complex factor \(\lambda (\alpha,F)\) such that \(z\to \lambda (\alpha,F) F(z)\) is analytically conjugated to the rotation by \(\alpha\). When \(F\) preserves the circle and has rotation number \(\alpha\), \(\lambda (\alpha,F)\) must be 1. One further studies the dependence of \(\lambda (\alpha,F)\) on \(F\) (it is holomorphic) and \(\alpha\) (this map is \(C^\infty\) in the sense of Whitney when restricted to the set of Brjuno rotation numbers). In general, one can define a complex rotation number for an annulus map and ask about sets of diffeomorphisms with a given rotation number. When this complex number is satisfied an arithmetic condition analogous to Brjuno’s, this set is a complex submanifold. The results are applied to the study of Herman rings in families of rational maps of the sphere. The set of maps which possess an Herman ring close to a given one for the initial map and with the same rotation number of Brjuno type is locally a codimension one complex set.
Proofs are obtained by a refinement of the geometric technique of Yoccoz applied in his solution of Siegel’s problem. The setting of the current paper causes added difficulties leading into the theory of several complex variables.

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37N05 Dynamical systems in classical and celestial mechanics
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References:

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