Wandering dynamics of transcendental functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914319640297472 |
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| author | Evdoridou, Vasiliki Martí-Pete, David Rempe, Lasse |
| author_facet | Evdoridou, Vasiliki Martí-Pete, David Rempe, Lasse |
| contents | We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$.
More precisely, let $φ$ be a holomorphic function on an open subset of the complex plane, and suppose that $K$ is a compact set such that $φ$ and all its iterates $φ^n$ are defined on $K$, and $φ^n(K)\to\infty$ as $n\to\infty$. We prove that there exist a transcendental meromorphic function $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ and a compact set $\widetilde{K}$ such that the dynamics of $f$ on the orbit of $\widetilde{K}$ is conjugate, via a smooth change of coordinate close to the identity, to that of $φ$ on the orbit of $K$. If $K$ does not separate the plane, the function $f$ may be chosen to be entire. If all iterates of $φ$ are univalent on $K$, we can take $\widetilde{K}=K$.
We also prove a similar theorem for oscillating dynamics. Finally, we use our results to answer a number of questions of Benini et al. concerning wandering domains of entire functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09952 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wandering dynamics of transcendental functions Evdoridou, Vasiliki Martí-Pete, David Rempe, Lasse Dynamical Systems Complex Variables Primary 37F10, Secondary 30D05, 37B45, 37B55, 37F20 We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $φ$ be a holomorphic function on an open subset of the complex plane, and suppose that $K$ is a compact set such that $φ$ and all its iterates $φ^n$ are defined on $K$, and $φ^n(K)\to\infty$ as $n\to\infty$. We prove that there exist a transcendental meromorphic function $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ and a compact set $\widetilde{K}$ such that the dynamics of $f$ on the orbit of $\widetilde{K}$ is conjugate, via a smooth change of coordinate close to the identity, to that of $φ$ on the orbit of $K$. If $K$ does not separate the plane, the function $f$ may be chosen to be entire. If all iterates of $φ$ are univalent on $K$, we can take $\widetilde{K}=K$. We also prove a similar theorem for oscillating dynamics. Finally, we use our results to answer a number of questions of Benini et al. concerning wandering domains of entire functions. |
| title | Wandering dynamics of transcendental functions |
| topic | Dynamical Systems Complex Variables Primary 37F10, Secondary 30D05, 37B45, 37B55, 37F20 |
| url | https://arxiv.org/abs/2602.09952 |