Real analytic nonexpansive maps on polyhedral normed spaces

TL;DR

This paper proves structural properties of real analytic nonexpansive maps on polyhedral normed spaces, including fixed point set isometries and periodic orbit bounds, confirming a special case of Nussbaum's conjecture.

Contribution

It establishes that fixed point sets are isometric to affine subspaces and confirms Nussbaum's $2^n$ Conjecture for real analytic $ orm{ullet}_ ext{1}$ and $ orm{ullet}_ ext{ extinfty}$ nonexpansive maps.

Findings

Fixed point sets are isometric to affine subspaces.

Periodic orbit periods divide a specific integer related to permutations.

Confirmed Nussbaum's $2^n$ Conjecture for certain real analytic maps.

Abstract

If a real analytic nonexpansive map on a polyhedral normed space has a nonempty fixed point set, then we show that there is an isometry from an affine subspace onto the fixed point set. As a corollary, we prove that for any real analytic 1-norm or $\infty$-norm nonexpansive map on $\mathbb{R}^n$, there is a positive integer $q$ such that the period of any periodic orbit divides $q$ and $q$ is the order, or twice the order, of a permutation on $n$ letters. This confirms Nussbaum's $2^n$ Conjecture for $\infty$-norm nonexpansive maps in the special case where the maps are also real analytic.