Non-Archimedean indifferent components of rational functions that are not disks

Nopal-Coello, Víctor

doi:10.1090/tran/8741

Non-Archimedean indifferent components of rational functions that are not disks
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by Víctor Nopal-Coello;

Trans. Amer. Math. Soc. 376 (2023), 419-451

DOI: https://doi.org/10.1090/tran/8741

Published electronically: September 2, 2022

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Abstract:

Consider a rational function of degree $d\geq 2$ acting on the Berkovich projective line of a complete and algebraically closed non-archimedean field. Rivera-Letelier has asked if a rational function in this setting can have infinitely many cycles of indifferent components that are not disks, and if not, if there exists a bound in terms of the degree of the function. In this work we show that for $d=2$ the bound is $d-1$. By imposing an extra condition on the residue field and a connectivity requirement on the cycles of indifferent components that are not disks, the bound $d-1$ is also achieved when $d\geq 3$. To ensure that the bound is realized, we describe how to construct a rational function of degree $d\geq 2$ with exactly $d-1$ cycles of indifferent components that are not disks.

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