On the boundary of an immediate attracting basin of a hyperbolic entire function
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908448424198144 |
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| author | Bergweiler, Walter Ding, Jie |
| author_facet | Bergweiler, Walter Ding, Jie |
| contents | Let $f$ be a transcendental entire function of finite order which has an attracting periodic point $z_0$ of period at least $2$. Suppose that the set of singularities of the inverse of $f$ is finite and contained in the component $U$ of the Fatou set that contains $z_0$. Under an additional hypothesis we show that the intersection of $\partial U$ with the escaping set of $f$ has Hausdorff dimension $1$. The additional hypothesis is satisfied for example if $f$ has the form $f(z)=\int_0^z p(t)e^{q(t)}dt+c$ with polynomials $p$ and $q$ and a constant $c$. This generalizes a result of Barański, Karpińska and Zdunik dealing with the case $f(z)=λe^z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the boundary of an immediate attracting basin of a hyperbolic entire function Bergweiler, Walter Ding, Jie Dynamical Systems Complex Variables 37F35, 37F10, 30D05 Let $f$ be a transcendental entire function of finite order which has an attracting periodic point $z_0$ of period at least $2$. Suppose that the set of singularities of the inverse of $f$ is finite and contained in the component $U$ of the Fatou set that contains $z_0$. Under an additional hypothesis we show that the intersection of $\partial U$ with the escaping set of $f$ has Hausdorff dimension $1$. The additional hypothesis is satisfied for example if $f$ has the form $f(z)=\int_0^z p(t)e^{q(t)}dt+c$ with polynomials $p$ and $q$ and a constant $c$. This generalizes a result of Barański, Karpińska and Zdunik dealing with the case $f(z)=λe^z$. |
| title | On the boundary of an immediate attracting basin of a hyperbolic entire function |
| topic | Dynamical Systems Complex Variables 37F35, 37F10, 30D05 |
| url | https://arxiv.org/abs/2407.19963 |