Power bounded composition operators on spaces of analytic functions

We study the dynamical behaviour of composition operators defined on spaces of real analytic functions. We characterize when such operators are power bounded, i.e. when the orbits of all the elements are bounded. In this case this condition is equivalent to the composition operator being mean ergodic. In particular, we show that the composition operator is power bounded on the space of real analytic functions on Omega if and only if there is a basis of complex neighbourhoods U of Omega such that the operator is an endomorphism on the space of holomorphic functions on each U.