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arXiv · 2211.10571

Preperiodic points with local rationality conditions in the quadratic unicritical family

Chatchai Noytaptim
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Abstract

For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic polynomials $f_c$ with rational parameters $c\in\mathbb{Q}$ so that $f_c$ has only finitely many totally real (totally $p$-adic, respectively) preperiodic points. These results rely on an adelic Fekete-type theorem and dynamics of the filled Julia set of $f_c$. Moreover, using a numerical criterion introduced in [NP], we make explicit calculations of the set of totally real $f_c$-preperiodic points when $c=-1,0,\frac{1}{5}$ and $\frac{1}{4}.$

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Chatchai Noytaptim. 2022-11-19. Preperiodic points with local rationality conditions in the quadratic unicritical family. https://arxiv.org/abs/2211.10571

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