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arXiv · 2305.09614

Algebraic periodic points of transcendental entire functions

David Krumm·Diego Marques·Carlos Gustavo Moreira·Pavel Trojovský
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Abstract

We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of $k$-periodic algebraic orbits, for all $k\geq 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.

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David Krumm, Diego Marques, Carlos Gustavo Moreira, Pavel Trojovský. 2023-05-16. Algebraic periodic points of transcendental entire functions. https://arxiv.org/abs/2305.09614

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