Linear Dynamics of Multiplication and Composition Operators on Hol(D).

We give a complete description of the linear dynamics of multiplication M m and composition operators C φ on the space Hol (D) of all holomorphic maps on the unit disc. We show that M m is never supercyclic, and cyclic if and only if the map m is injective. For composition operators, we prove that if φ has a fixed point in D , then C φ is either not cyclic, or cyclic but not supercyclic on Hol (D) . On the other hand, if φ does not have any fixed point in the unit disc, then C φ is hypercyclic on Hol (D) . We provide explicit expressions of cyclic and hypercyclic vectors. Finally, we make some observations on weighted composition operators on Hol (D) . [ABSTRACT FROM AUTHOR]