A counterexample to parabolic dichotomies in holomorphic iteration
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arXiv
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| Format: | Preprint |
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2023
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| author | Arosio, Leandro Bracci, Filippo Gaussier, Herv/'e |
| author_facet | Arosio, Leandro Bracci, Filippo Gaussier, Herv/'e |
| contents | We give an example of a parabolic holomorphic self-map $f$ of the unit ball $\mathbb B^2\subset \mathbb C^2$ whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc $\mathbb D\subset \mathbb C$, which can be chosen to be different from the identity. As a consequence, in contrast to the one dimensional case, this provides a first example of a holomorphic self-map of the unit ball which has points with zero hyperbolic step and points with nonzero hyperbolic step, solving an open question and showing that parabolic dynamics in the ball $\\mathbb B^2$ is radically different from parabolic dynamics in the disc. The example is obtained via a geometric method, embedding the ball $\mathbb B^2$ as a domain $Ω$ in the bidisc $\\mathbb D\times \mathbb{H}$ that is forward invariant and absorbing for the map $(z,w)\mapsto (e^{iθ}z,w+1)$, where $\mathbb H\subset \mathbb C$ denotes the right half-plane. We also show that a complete Kobayashi hyperbolic domain $Ω$ with such properties cannot be Gromov hyperbolic w.r.t. the Kobayashi distance (hence, it cannot be biholomorphic to $\\mathbb B^2$) if an additional quantitative geometric condition is satisfied. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_15739 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A counterexample to parabolic dichotomies in holomorphic iteration Arosio, Leandro Bracci, Filippo Gaussier, Herv/'e Complex Variables Dynamical Systems Metric Geometry 32H50, 32F45, 53C23 We give an example of a parabolic holomorphic self-map $f$ of the unit ball $\mathbb B^2\subset \mathbb C^2$ whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc $\mathbb D\subset \mathbb C$, which can be chosen to be different from the identity. As a consequence, in contrast to the one dimensional case, this provides a first example of a holomorphic self-map of the unit ball which has points with zero hyperbolic step and points with nonzero hyperbolic step, solving an open question and showing that parabolic dynamics in the ball $\\mathbb B^2$ is radically different from parabolic dynamics in the disc. The example is obtained via a geometric method, embedding the ball $\mathbb B^2$ as a domain $Ω$ in the bidisc $\\mathbb D\times \mathbb{H}$ that is forward invariant and absorbing for the map $(z,w)\mapsto (e^{iθ}z,w+1)$, where $\mathbb H\subset \mathbb C$ denotes the right half-plane. We also show that a complete Kobayashi hyperbolic domain $Ω$ with such properties cannot be Gromov hyperbolic w.r.t. the Kobayashi distance (hence, it cannot be biholomorphic to $\\mathbb B^2$) if an additional quantitative geometric condition is satisfied. |
| title | A counterexample to parabolic dichotomies in holomorphic iteration |
| topic | Complex Variables Dynamical Systems Metric Geometry 32H50, 32F45, 53C23 |
| url | https://arxiv.org/abs/2310.15739 |