Iterates of composition operators on global spaces of ultradifferentiable functions

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 $\Sigma_{d'}.$