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 S(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 φ 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 φ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.