Non linearizable holomorphic dynamics having an uncountable number of symmetries
Let \(\text{Diff} (\mathbb{C},0)\) be the group of germs of holomorphic diffeomorphisms of \((\mathbb{C},0)\). The centralizer of \(f\in \text{Diff} (\mathbb{C},0)\), \(\text{Cent} (f)\), is the subgroup of \(\text{Diff} (\mathbb{C},0)\) of elements commuting with \(f\). A holomorphic germ \(f(z) = \lambda z + O(z^2)\), \(\lambda \in\mathbb{C}^*\), is linearizable if it is holomorphically conjugated to its linear part. Depending on \(\lambda\) more or less is known about the group \(\text{Cent} (f)\). It is well understood the case when \(|\lambda |= |Df(0) |\neq 1\) and the case when \(|\lambda |= 1\) and \(\lambda\) is a root of unity. The paper under review deals with the irrational case \(\lambda = Df(0) = \exp^{2 \pi i \alpha}\), \(\alpha \in\mathbb{R} \backslash\mathbb{Q}\), and \(f\) nonlinearizable. It is proved that there exists a nonlinearizable holomorphic diffeomorphism of \((\mathbb{C},0)\) having an uncountable centralizer. Thus it is given a positive answer to a problem posed by M. R. Herman. The author constructs a family of holomorphic germs indexed by rotation numbers in an explicit Cantor set \(K \subset\mathbb{R}/\mathbb{Z}\), \((f_t)_{t \in K}\), \(f_t(z) = \exp^{2 \pi it} z+ O(z^2)\), univalent on the unit disk, and nonlinearizable for \(t\in K \cap (\mathbb{R} \backslash\mathbb{Q})/\mathbb{Z}\). The maps of the family have an infinite number of finite order symmetries. The family \((f_t)_{t \in K \cap\mathbb{Q}/Z}\) has no common periodic orbit (except 0). It is also constructed a similar family having a sequence of common periodic orbits with 0 as a limit point. A detailed description of the construction is given. The author constructs nonlinearizable holomorphic germs developing geometrical techniques, based on a method invented by J. C. Yoccoz. The second part of the paper deals with the study of centralizers of analytic diffeomorphisms of the circle. There are constructed nonlinearizable analytic diffeomorphisms of the circle having an irrational rotation number and uncountable centralizer. The construction is derived from the above results and using a tool due to the author which relates analytic diffeomorphisms of the circle, and indifferent fixed points for holomorphic maps.
scientific article; zbMATH DE number 763282
scientific article