Mean and pointwise ergodicity for composition operators on rearrangement-invariant spaces

Abstract

We study ergodicity of composition operators on rearrangement-invariant Banach function spaces. More precisely, we give a natural and easy-to-check condition on the symbol of the operator which entails mean ergodicity on a very large class of rearrangement-invariant Banach function spaces. Further, we present some natural additional assumptions that allow us to obtain pointwise ergodicity. The class of spaces covered by our results contains many non-reflexive spaces, such as the Lorentz spaces $L^{p, 1}$ and $L^{p,\infty}$, $p \in (1, \infty)$, Orlicz spaces $L \log^{\alpha} L$ and $\exp L^{\alpha}$, $\alpha > 0$, and the spaces $L^1$ and $L^{\infty}$ over measure spaces of finite measure. The main novelty in our approach is the application of a new locally convex topology which we introduce and which lies strictly between the norm topology and the weak topology induced by the associate space. Throughout, we give several examples which illustrate the applicability of our results as well as highlight the necessity and optimality of our assumptions.