A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting

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.