Greenś Mapping and Julia Sets
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913977311690752 |
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| author | Binder, Ilia Müller, Paul F. X. Yuditskii, Peter |
| author_facet | Binder, Ilia Müller, Paul F. X. Yuditskii, Peter |
| contents | In March 1999, the first named author (Binder) posed the problem of showing that a ``good direction'' $ψ\in [0,2]$ exists, for any Green's mapping $T:H\rightarrow\tilde Ω$, i.e., \begin{equation}\label{binder}
\int\limits_0\limits^{1} |T''(re^{iπψ})|dr <\infty, \quad\text{ for at least one }
\quad ψ\in [0,2]. \end{equation} Presently this problem is open even in the special case where $\partial Ω$ is a uniformly perfect subset of the real line. In this paper we obtain a positive solution when $Ω= \overline{C} \setminus E_0$ where $E_0 \subset R $ is the Julia set of an expanding quadratic polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Greenś Mapping and Julia Sets Binder, Ilia Müller, Paul F. X. Yuditskii, Peter Complex Variables Dynamical Systems In March 1999, the first named author (Binder) posed the problem of showing that a ``good direction'' $ψ\in [0,2]$ exists, for any Green's mapping $T:H\rightarrow\tilde Ω$, i.e., \begin{equation}\label{binder} \int\limits_0\limits^{1} |T''(re^{iπψ})|dr <\infty, \quad\text{ for at least one } \quad ψ\in [0,2]. \end{equation} Presently this problem is open even in the special case where $\partial Ω$ is a uniformly perfect subset of the real line. In this paper we obtain a positive solution when $Ω= \overline{C} \setminus E_0$ where $E_0 \subset R $ is the Julia set of an expanding quadratic polynomial. |
| title | Greenś Mapping and Julia Sets |
| topic | Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2508.04207 |