Stronger Forms of Transitivity and Sensitivity for Nonautonomous Discrete Dynamical Systems and Furstenberg Families

Let (Y, d) be a nontrivial metric space and (Y, g(1,infinity)) be a nonautonomous discrete dynamical system given by sequences (gl)l=1 infinity and F2 be given shift-invariant Furstenberg families. In this paper, we study stronger forms of transitivity and sensitivity for nonautonomous discrete dynamical systems by using Furstenberg family. In particular, we discuss the F-transitivity, F-mixing, F-sensitivity, F-collective sensitivity, F-synchronous sensitivity, (F-1, F-2)-sensitivity and F-multi-sensitivity for the system (Y, g(1,infinity)) and show that under the conditions that gj is semi-open and satisfies g(j) o g = g o gj for each j. {1, 2, ...} and that (, 1,) is -transitive if and only if so is (, ). Yg infinity Yg(, 1,) is -mixing if and only if so is (, ). Yg infinity Yg(, 1,) is -sensitive if and only if so is (, ). Yg infinity Yg(, 1,) is -sensitive if and only if so is (, ). Yg infinity Yg(, 1,) is -collectively sensitive if and only if so is (, ). Yg infinity Yg(, 1,) is -synchronous sensitive if and only if so is (, ). Yg infinity Yg(, 1,) is -multi-sensitive if and only if so is (, ). The above results extend the existing ones.