Fixed Points of the q-Bracket on the p-Adic Unit Disk

TL;DR

This paper investigates the fixed points of the q-bracket function on the p-adic unit disk, revealing manifold structures, an analytic contraction function, and special cases for p=3.

Contribution

It characterizes the fixed points of the q-bracket on p-adic disks, introduces an analytic contraction function, and analyzes the structure over Z_p.

Findings

The fixed points form a manifold with degree p-2 projections.

An analytic function Q(X) maps x to q with fixed points.

The theory is trivial over Z_p unless p=3.

Abstract

We study the fixed points of the q-bracket on the complex unit disk, and prove the following. The set of (nontrivial) pairs (x,q) such that [x]_q=x form a manifold whose standard projections both have degree p-2. There is an analytic function Q(X) taking x to q for which [x]_q=x, which is a (bijective) contraction unless the multiplicity of the residue of x in the fiber over q is two. The restriction of the theory to Z_p is trivial unless p=3.