  We study the spectrum of operators in the Schwartz space of rapidly
decreasing functions which associate each function with its composition with a
polynomial. In the case where this operator is mean ergodic we prove that its
spectrum reduces to 0, while the spectrum of any non mean ergodic composition
operator with a polynomial always contains the closed unit disc except perhaps
the origen. We obtain a complete description of the spectrum of the composition
operator with a quadratic polynomial or a cubic polynomial with positive
leading coefficient.
