Dynamics and spectra of composition operators on the Schwartz space

In this paper we study the dynamics of the composition operators defined in the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol $\varphi$ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when $\varphi$ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.