标题： 关于Borsuk-Ulam情形的摄动多重性定理 显示英文标题

标题： A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting

摘要： 我们证明了在球面的小扰动（摇晃）下，经典Borsuk--Ulam定理的一个推广。 我们表明，对于连续映射$f : S^2 \to \mathbb{R}^2$的一般扰动，使得$f_\epsilon(x) = f_\epsilon(-x)$成为有限且奇数的点$x \in S^2$的数量可能超过一个对径重合的经典下界。 特别是，我们展示了存在具有3、5或7个这样的点的映射，并解释了在更高复杂度的扰动下这种多重性的无界性质。 显示英文摘要

摘要： 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.