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

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.

11 pages. The new version (7/11) has an expanded introduction and some sharper results now contained in Theorem 6, including a criterion for when the map Q(X) is a contraction (the multiplicity of the residue of x should be 1). There is no more Proposition 7 and 8. There are also slight notation changes throughout, to enhance the exposition