A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting

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.