Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces
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| Format: | Preprint |
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2022
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| author | Arosio, Leandro Fiacchi, Matteo Guerini, Lorenzo Karlsson, Anders |
| author_facet | Arosio, Leandro Fiacchi, Matteo Guerini, Lorenzo Karlsson, Anders |
| contents | We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $Ω\subset\subset \mathbb{C}^q$, where $Ω$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_17480 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces Arosio, Leandro Fiacchi, Matteo Guerini, Lorenzo Karlsson, Anders Complex Variables Dynamical Systems Metric Geometry Primary 32H50, Secondary 32F45, 53C23 We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $Ω\subset\subset \mathbb{C}^q$, where $Ω$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary. |
| title | Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces |
| topic | Complex Variables Dynamical Systems Metric Geometry Primary 32H50, Secondary 32F45, 53C23 |
| url | https://arxiv.org/abs/2210.17480 |