Furstenberg family and multi-transitivity in non-autonomous systems

In this paper, we study the dynamics of a non-autonomous dynamical system (X, f(1,infinity)) generated by a sequence (f(n)) of continuous self maps. We obtain necessary and sufficient condition for a non-autonomous system to be multi-F-transitive and F-mixing of all orders where F is a Furstenberg family. Also, we get some sufficient conditions under which multi-F-transitivity of a non-autonomous discrete dynamical system is inherited under iterations. We relate the multi-F-transitivity of the non-autonomous system (X,f(1,infinity)) with the multi-F-transitivity of (X, f). We also show that multi-F-transitivity is equivalent with F-mixing of all orders if the system is minimal. We also give examples to investigate the conditions imposed for the results to hold good.