Notions of mixing and sensitivities for triangular map and its non-autonomous components

In this paper, we relate the dynamics of the triangular map to the dynamics of its individual components. We prove that if the non-autonomous system generated by a transitive point (for the base map) is topological mixing then the triangular map is transitive. We prove that if the base map is minimal and the generating family of non-autonomous systems is commutative then weak mixing of the non-autonomous system (generated by a transitive point) ensures transitivity of the triangular system. We also derive sufficient conditions for triangular map to exhibit stronger notions of mixing and provide examples to establish the necessity of the conditions imposed. We prove that a triangular system is equicontinuous if and only if each of the component systems are equicontinuous and the non-autonomous components { ( Y , Dx ) : x ∈ X } are synchronized. We prove that if the family of non-autonomous systems is synchronized then if non-autonomous system generated by a transitive point exhibits any form of mixing then non-autonomous system generated by any point exhibits the same. We also relate various forms of sensitivities for triangular map to analogous notions for the component systems.