Arboreal Galois representations and uniformization of polynomial dynamics

Abstract

Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory.

References in corpus (3)

• p-adic logarithms for polynomial dynamics
• Uniform Bounds on Pre-Images under Quadratic Dynamical Systems
• Rational preimages in families of dynamical systems

Cited by in corpus (7)

• Local arboreal representations
• The arithmetic basilica: a quadratic PCF arboreal Galois group
• Arboreal Cantor actions
• Galois uniformity in quadratic dynamics over rational function fields
• Bounded height in families of dynamical systems
• Böttcher coordinates at wild superattracting fixed points
• p-adic uniformization and the action of Galois on certain affine correspondences