Interactions between universal composition operators and complex dynamics

This paper is concerned with universality properties of composition operators $C_f$, where the symbol $f$ is given by a transcendental entire function restricted to parts of its Fatou set. We determine universality of $C_f$ when $f$ is restricted to (subsets of) Baker and wandering domains. We then describe the behaviour of universal vectors, under the action of iterates of the symbol $f$, near periodic points of $f$ or near infinity. Finally, we establish a principal universality theorem for the more general class of weighted composition operators, which we then apply to uncover universality results in the context of various types of Fatou components of the associated symbol.