  We analyze the behavior of the iterates of composition operators defined by
polynomials acting on global classes of ultradifferentiable functions of
Beurling type and being invariant under Fourier transform. We characterize the
polynomials $\psi$ for which the sequence of iterates is equicontinuous between
two different Gelfand-Shilov spaces. For the particular case in which the
weight $\omega$ is equivalent to a power of the logarithm, the result obtained
characterizes the polynomials $\psi$ for which the composition operator
$C_\psi$ is power bounded in ${\mathcal S}_\omega({\mathbb R}).$ Unlike the
composition operators in Schwartz class, the Waelbroek spectrum of an operator
$C_\psi$, being $\psi$ a polynomial of degree greater than one lacking fixed
points is never compact. We focus on the problem of convergence of Neumann
series. We deduce the continuity of the resolvent operator between two
different Gelfand-Shilov classes for polynomials $\psi$ lacking fixed points.
Concerning polynomials of second degree the most interesting case is the one in
which the polynomial only has one fixed point: we provide some restrictions on
the indices $d, d'$ that are necessary for the resolvent operator to be
continuous between the Gelfand-Shilov classes $\Sigma_d$ and $\Sigma_{d'}.$
