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Fixed-point proportion of geometric iterated Galois groups

Jorge Fariña-Asategui, Santiago Radi

TL;DR

The paper resolves the problem of determining the fixed-point proportion of geometric iterated Galois groups, proving that only Chebyshev-type polynomials yield positive $\mathrm{FPP}$; it develops a unified framework using mixing self-similar groups, martingale methods, and ergodic theory, and extends the analysis to rational functions via exceptional sets and Thurston orbifolds. The main technical innovation is a commutator trick that, combined with a pseudomixing property, shows that mixing groups have $\mathrm{FPP}=0$ except in Euclidean orbifold cases. A complete classification is obtained for polynomials, linking positive $\mathrm{FPP}$ to Euclidean orbifolds of type $(2,2,\infty)$ and, over separably closed fields, twisted Chebyshev polynomials. These results unify and extend prior partial findings and have direct implications for the distribution of periodic points over finite fields and related Arakelov-type dynamical problems.

Abstract

In 1980, Odoni initiated the study of the fixed-point proportion of iterated Galois groups of polynomials motivated by prime density problems in arithmetic dynamics. The main goal of the present paper is to completely settle the longstanding open problem of computing the fixed-point proportion of geometric iterated Galois groups of polynomials. Indeed, we confirm the well-known conjecture that Chebyshev polynomials are the only complex polynomials whose geometric iterated Galois groups have positive fixed-point proportion. Our proof relies on methods from group theory, ergodic theory, martingale theory and complex dynamics. This result has direct applications to the proportion of periodic points of polynomials over finite fields. The general framework developed in this paper applies more generally to rational functions over arbitrary fields and generalizes, via a unified approach, previous partial results, which have all been proved with very different methods.

Fixed-point proportion of geometric iterated Galois groups

TL;DR

The paper resolves the problem of determining the fixed-point proportion of geometric iterated Galois groups, proving that only Chebyshev-type polynomials yield positive ; it develops a unified framework using mixing self-similar groups, martingale methods, and ergodic theory, and extends the analysis to rational functions via exceptional sets and Thurston orbifolds. The main technical innovation is a commutator trick that, combined with a pseudomixing property, shows that mixing groups have except in Euclidean orbifold cases. A complete classification is obtained for polynomials, linking positive to Euclidean orbifolds of type and, over separably closed fields, twisted Chebyshev polynomials. These results unify and extend prior partial findings and have direct implications for the distribution of periodic points over finite fields and related Arakelov-type dynamical problems.

Abstract

In 1980, Odoni initiated the study of the fixed-point proportion of iterated Galois groups of polynomials motivated by prime density problems in arithmetic dynamics. The main goal of the present paper is to completely settle the longstanding open problem of computing the fixed-point proportion of geometric iterated Galois groups of polynomials. Indeed, we confirm the well-known conjecture that Chebyshev polynomials are the only complex polynomials whose geometric iterated Galois groups have positive fixed-point proportion. Our proof relies on methods from group theory, ergodic theory, martingale theory and complex dynamics. This result has direct applications to the proportion of periodic points of polynomials over finite fields. The general framework developed in this paper applies more generally to rational functions over arbitrary fields and generalizes, via a unified approach, previous partial results, which have all been proved with very different methods.
Paper Structure (20 sections, 23 theorems, 148 equations, 1 figure)

This paper contains 20 sections, 23 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Groups acting on trees
  3. Self-similar and fractal groups
  4. Mixing groups
  5. Connection to projection-invariant subgroups
  6. Dynamical systems associated to mixing groups
  7. Dynamical systems associated to fractal groups
  8. A pseudomixing property
  9. Fixed-point proportion and the martingale strategy
  10. Fixed-point processes, martingales and the fixed-point proportion
  11. Proof of the main result
  12. Geometric iterated Galois groups of rational functions
  13. Geometric iterated Galois groups
  14. Exceptional sets and generalizations
  15. Commutator trick
  16. ...and 5 more sections

Key Result

Theorem 2

Let $K$ be a field and $f\in K[x]$ a polynomial of degree $d \geq 2$, such that either $\mathrm{char}(K)=0$ or $\mathrm{char}(K)$ does not divide the local degree of any critical point of $f$. Then, for $t$ transcendental over $K$, either

Figures (1)

  • Figure 1: A graphical representation of the $f$-orbit of $p\in P_f\setminus \Delta_f$ in the proof of \ref{['theorem: good generators are seen']}, where $z\in f^{-1}(p)$.

Theorems & Definitions (49)

  • Conjecture 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Conjecture 7
  • Remark 2.1
  • Definition 2.2: Mixing group
  • Lemma 2.3: Commutator trick
  • ...and 39 more