Fixed-point proportion of geometric iterated Galois groups

TL;DR

This paper proves that Chebyshev polynomials are uniquely characterized by having positive fixed-point proportions in their geometric iterated Galois groups, resolving a longstanding open problem in arithmetic dynamics.

Contribution

It provides a complete solution to the open problem of computing fixed-point proportions of geometric iterated Galois groups, confirming a key conjecture about Chebyshev polynomials.

Findings

Chebyshev polynomials are the only complex polynomials with positive fixed-point proportion in their geometric iterated Galois groups.

The proof combines group theory, ergodic theory, martingale theory, and complex dynamics.

Applications include understanding the distribution of periodic points over finite fields.

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.