Mixing, simultaneous universal and disjoint universal backward Φ-shifts

Spaces of sequences which are valued in a topological space E are considered in order to study universality properties of generalized backward shifts associated to certain selfmappings of E. After providing a condition which guarantees the mixing property of backward Φ-shifts in terms of the dynamical properties of the generating selfmappings, applications to universality theory within the framework of complex dynamics are furnished.