  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.
