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On nonlinearizable holomorphic dynamics and a conjecture of V. I. Arnold. (Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnold.) (French) Zbl 0812.58051

This extensive and respectable paper deals, among other things, with Arnold’s conjecture concerning the existence of a non-linearizable analytic circle diffeomorphism for which any analytic extension has not any periodic orbit. In order to verify this conjecture the author proves
Theorem 1. There exists a circle diffeomorphism \(f\) analytic on some neighbourhood \(U\) of the circle such that the following holds:
– \(f\) has irrational rotation number
– If for some \(z \in U\) the iterates \(f^{\circ n}(z)\) are defined and contained in \(U\) for all \(n \in N\) then the sequence of iterates has at least one accumulation point on the circle.
– The common domain of holomorphy of \(f\) and its iterates \(f^{\circ n}\), where \(n \in \mathbb{N}\), equals the component of the set of all points lying in \(U\) such that all iterates are well defined and contained in \(U\) which contains the circle.
Concerning holomorphic germs the author establishes
Theorem 2. For \(\alpha \in \mathbb{R} \setminus \mathbb{Q}\) satisfying the diophantine condition \(\sum_{k \in \mathbb{N}} \log \log q_{k + 1} / q_ k = \infty\) there exists some \(f : z \mapsto e^{2\pi i \alpha} z + \dots\) holomorphic and injective on the unit disc \(\mathbb{D}\) such that for every \(z \in \mathbb{D}\) the sequence of the iterates accumulates at 0 provided all iterates are defined and contained in \(\mathbb{D}\).
and shows this result to be optimal.
Theorem 3. If \(\alpha \in \mathbb{R} \setminus \mathbb{Q}\) satisfies \(\sum_{k \in \mathbb{N}} \log \log q_{k + 1}/q_ k < \infty\) then every non-linearizable mapping \(f: z \mapsto e^{2 \pi i\alpha} z + \dots\) holomorphic and injective on \(\mathbb{D}\) has a sequence of periodic points \(z_ n\) converging to 0 such that the sequence of the corresponding periods \(p_ n\) satisfies the Brjuno-condition \(\sum_{n \in \mathbb{N}} \log p_{n + 1}/p_ n = \infty\).
Using this result the author generalizes a result of Yoccoz.
Theorem 4. Let \(\lambda = e^{2 \pi i\alpha}\) not satisfy the Brjuno condition. Then every polynomial \(p(z) = \lambda z + \dots\) structurally stable has a sequence of periodic points converging to 0 and therefore is non-linearizable.
This clear and carefully written paper bases on methods and results due to Yoccoz and the author.
Reviewer: H.Kriete (Aachen)

MSC:

37B99 Topological dynamics
26A18 Iteration of real functions in one variable
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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