The generalized cyclotomic mappings over finite fields $\mathbb{F}_{q}$ are those mappings which induce monomial functions on all cosets of an index $\ell$ subgroup $C_0$ of the multiplicative group $\mathbb{F}_{q}^{*}$. Previous research has focused on the one-to-one property, the functional graphs, and their applications in constructing linear codes and bent functions. In this paper, we devote to study the many-to-one property of these mappings. We completely characterize many-to-one generalized cyclotomic mappings for $1 \le \ell \le 3$. Moreover, we completely classify $2$-to-$1$ generalized cyclotomic mappings for any divisor $\ell$ of $q-1$. In addition, we construct several classes of many-to-one binomials and trinomials of the form $x^r h(x^{q-1})$ on $\mathbb{F}_{q^2}$, where $h(x)^{q-1}$ induces monomial functions on the cosets of a subgroup of $U_{q+1}$.

翻译：有限域 $\mathbb{F}_{q}$ 上的广义分圆映射是指那些在乘法群 $\mathbb{F}_{q}^{*}$ 的指数为 $\ell$ 的子群 $C_0$ 的所有陪集上诱导出单项式函数的映射。先前的研究主要集中在其一对一性质、函数图及其在构造线性码和 Bent 函数中的应用。本文致力于研究这些映射的多对一性质。我们完全刻画了当 $1 \le \ell \le 3$ 时的多对一广义分圆映射。此外，对于 $q-1$ 的任意除数 $\ell$，我们完全分类了 $2$-对-$1$ 的广义分圆映射。另外，我们在 $\mathbb{F}_{q^2}$ 上构造了若干类形如 $x^r h(x^{q-1})$ 的多对一二项式与三项式，其中 $h(x)^{q-1}$ 在 $U_{q+1}$ 的一个子群的陪集上诱导出单项式函数。