  The functional graph of a function $g:X\rightarrow X$ is the directed graph
with vertex set $X$ the edges of which are of the form $x\rightarrow g(x)$ for
$x\in X$. Functional graphs are heavily studied because they allow one to
understand the behavior of $g$ under iteration (i.e., to understand the
discrete dynamical system $(X,g)$), which has various applications, especially
when $X$ is a finite field $\mathbb{F}_q$. This paper is an extensive study of
the functional graphs of so-called index $d$ generalized cyclotomic mappings of
$\mathbb{F}_q$, which are a natural and manageable generalization of monomial
functions. We provide both theoretical results on the structure of their
functional graphs and Las Vegas algorithms for solving fundamental problems,
such as parametrizing the connected components of the functional graph by
representative vertices, or describing the structure of a connected component
given by a representative vertex. The complexity of these algorithms is
analyzed in detail, and we make the point that for fixed index $d$ and most
prime powers $q$ (in the sense of asymptotic density), suitable implementations
of these algorithms have an expected runtime that is polynomial in $\log{q}$ on
quantum computers, whereas their expected runtime is subexponential in
$\log{q}$ on a classical computer. We also discuss four special cases in which
one can devise Las Vegas algorithms with this kind of complexity behavior over
most finite fields that solve the graph isomorphism problem for functional
graphs of generalized cyclotomic mappings.
