On many-to-one property of generalized cyclotomic mappings

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 $U_{q+1}$.