Non linearizable holomorphic dynamics having an uncountable number of symmetries
type: Item
instance of: scholarly article
title: Non linearizable holomorphic dynamics having an uncountable number of symmetries
zbMATH Open document ID: 0862.58045
DOI: 10.1007/BF01245175
published in: Inventiones Mathematicae
publication date: 1997-05-25T00:00:00Z
full work available at URL: https://eudml.org/doc/144249
review text: 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 irr…
Mathematics Subject Classification ID: 37F99
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 763282
zbMATH Keywords: holomorphic diffeomorphisms
zbMATH Keywords: finite order symmetries
zbMATH Keywords: rotation number
author: Ricardo Pérez-Marco
reviewed by: Emilia Petrisor
MaRDI profile type: Publication
cites work: Über die Häufigkeit der Nichtzentren
cites work: On commuting circle mappings and simultaneous Diophantine approximations
cites work: A note on non-linearizable analitic functions
cites work: Regular iteration of real and complex functions
OpenAlex ID: W2022780924
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description: scientific article; zbMATH DE number 763282