Symmetrization and the rate of convergence of semigroups of holomorphic functions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916662903570432 |
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| author | Betsakos, Dimitrios Christodoulou, Argyrios |
| author_facet | Betsakos, Dimitrios Christodoulou, Argyrios |
| contents | Let $(ϕ_t)$, $t\ge 0$, be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$. Let $Ω$ be its Koenigs domain and $τ\in \partial \mathbb{D}$ be its Denjoy-Wolff point. Suppose that $0\in Ω$ and let $Ω^\sharp$ be the Steiner symmetrization of $Ω$ with respect to the real axis. Consider the semigroup $(ϕ_t^\sharp)$ with Koenigs domain $Ω^\sharp$ and let $τ^\sharp$ be its Denjoy-Wolff point. We show that, up to a multiplicative constant, the rate of convergence of $(ϕ_t^\sharp)$ is slower than that of $(ϕ_t)$; that is, for every $t>0$, $|ϕ_t(0)-τ|\leq 4π\, |ϕ_t^\sharp(0)-τ^\sharp|$. The main tool for the proof is the harmonic measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetrization and the rate of convergence of semigroups of holomorphic functions Betsakos, Dimitrios Christodoulou, Argyrios Complex Variables Dynamical Systems Let $(ϕ_t)$, $t\ge 0$, be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$. Let $Ω$ be its Koenigs domain and $τ\in \partial \mathbb{D}$ be its Denjoy-Wolff point. Suppose that $0\in Ω$ and let $Ω^\sharp$ be the Steiner symmetrization of $Ω$ with respect to the real axis. Consider the semigroup $(ϕ_t^\sharp)$ with Koenigs domain $Ω^\sharp$ and let $τ^\sharp$ be its Denjoy-Wolff point. We show that, up to a multiplicative constant, the rate of convergence of $(ϕ_t^\sharp)$ is slower than that of $(ϕ_t)$; that is, for every $t>0$, $|ϕ_t(0)-τ|\leq 4π\, |ϕ_t^\sharp(0)-τ^\sharp|$. The main tool for the proof is the harmonic measure. |
| title | Symmetrization and the rate of convergence of semigroups of holomorphic functions |
| topic | Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2503.20447 |