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Structure of the Components of the Fixed Locus of a Self-Map of the Berkovich Line

Xander Faber, Niladri Patra

Abstract

We describe the local and global structure of the fixed locus for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. This includes a bound for the number of connected components that is sharp when the residue characteristic is large or zero. The case of small nonzero residue characteristic will be treated in a subsequent article.

Structure of the Components of the Fixed Locus of a Self-Map of the Berkovich Line

Abstract

We describe the local and global structure of the fixed locus for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. This includes a bound for the number of connected components that is sharp when the residue characteristic is large or zero. The case of small nonzero residue characteristic will be treated in a subsequent article.

Paper Structure

This paper contains 13 sections, 29 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Notation
  4. Preliminaries
  5. Connected subsets of the analytic line
  6. Connected components of the fixed locus
  7. Surplus multiplicities and fixed points
  8. Indifference lemmas
  9. Invertible maps
  10. Indifferent components
  11. Classical fixed points in a non-classical component
  12. Hyperbolic components
  13. Crucial points and fixed points

Key Result

Theorem 1.3

If $f \in K(z)$ is a rational function of degree $d \ge 1$, then

Figures (1)

  • Figure 1: Schematic of the id-indifference locus (red) and additively indifferent points (blue) of ${\mathrm{Fix}}(f)$. A general additively indifferent point has been labeled in each figure. The hyperbolic lengths of the maximal fixed arcs leading off of $[0,\infty]$ are discussed at the end of this section.

Theorems & Definitions (63)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3: Rumely's Weight Formula Rumely_new_equivariant
  • Corollary 1.4
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['prop:closest_point']}
  • Proposition 2.3
  • proof
  • ...and 53 more