A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting

TL;DR

This paper extends the Borsuk-Ulam theorem to perturbed spheres, showing that generic small changes lead to a finite, odd number of antipodal coincidence points, revealing new multiplicity phenomena.

Contribution

It introduces a perturbative version of the Borsuk-Ulam theorem, demonstrating how small perturbations affect the number of antipodal points where the map coincides.

Findings

Number of antipodal coincidence points becomes finite and odd under perturbation

Existence of maps with 3, 5, or 7 such points

Multiplicity can be unbounded with increased perturbation complexity

Abstract

We prove a generalization of the classical Borsuk--Ulam Theorem under small perturbations (shaking) of the sphere. We show that for a generic perturbation of a continuous map $f : S^2 \to \mathbb{R}^2$, the number of points $x \in S^2$ such that $f_\epsilon(x) = f_\epsilon(-x)$ becomes finite and odd, and may exceed the classical lower bound of one antipodal coincidence. In particular, we show the existence of maps with 3, 5, or 7 such points, and explain the unbounded nature of this multiplicity under higher complexity of the perturbation.