On disjoint dynamical properties and Lipschitz-free spaces

The notion of disjoint A-transitivity for a Furstenberg family A is introduced with the aim to generalize properties derived from disjoint hypercyclic operators. We begin a systematic study by showing some of the basic properties, including necessary conditions to inherit the property on the whole space from an invariant linearly dense set containing the origin. As a consequence, we continue the study of the link between non-linear and linear dynamics through Lipschitz-free spaces by presenting some necessary conditions to obtain disjoint A-transitivity for families of Lipschitz-free operators on F (M) expressed in terms of conditions in the underlying metric space M.