Compressive Domains and a Bound for the Number of Components of the Fixed Locus of a Self-Map of the Berkovich Line

Xander Faber; Niladri Patra

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arXiv · 2608.14545

Compressive Domains and a Bound for the Number of Components of the Fixed Locus of a Self-Map of the Berkovich Line

Xander Faber·Niladri Patra

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Abstract

We introduce the notion of a "compressive domain" for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. We prove that such a domain always contains a classical fixed point, and we leverage this fact to give a sharp upper bound for the number of connected components of the fixed locus of a rational function. We give a second proof for polynomial functions that uses a previously unpublished mass formula of Rivera-Letelier. Finally, we give an explicit formula for the crucial weight inside a compressive domain as a function of the number of classical fixed points and boundary points.

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Xander Faber, Niladri Patra. 2026-08-14. Compressive Domains and a Bound for the Number of Components of the Fixed Locus of a Self-Map of the Berkovich Line. https://arxiv.org/abs/2608.14545

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