Fixed-point proportion of geometric iterated Galois groups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918300119728128 |
|---|---|
| author | Fariña-Asategui, Jorge Radi, Santiago |
| author_facet | Fariña-Asategui, Jorge Radi, Santiago |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16173 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fixed-point proportion of geometric iterated Galois groups Fariña-Asategui, Jorge Radi, Santiago Number Theory Dynamical Systems Group Theory Primary: 37P05, 20E08, 37A50, Secondary: 60G42, 37F10, 37P35 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. |
| title | Fixed-point proportion of geometric iterated Galois groups |
| topic | Number Theory Dynamical Systems Group Theory Primary: 37P05, 20E08, 37A50, Secondary: 60G42, 37F10, 37P35 |
| url | https://arxiv.org/abs/2601.16173 |