Some stronger forms of topological transitivity and sensitivity for a sequence of uniformly convergent continuous maps

Let ( H , d ) be a metric space, F be a Furstenberg family, and ( g m ) m ∈ Z + be a sequence of continuous map on H, which converges uniformly to a map g on H. In this paper, under the condition lim m → ∞ ⁡ d ∞ ( g m m , g m ) = 0 , a necessary and sufficient condition for g to be F -mixing is established. Moreover, let T ⊂ Z + be an infinite set and lim m ∈ T : m → ∞ ⁡ d ∞ ( g m m , g m ) = 0 . Then, some necessary and sufficient conditions, or sufficient conditions, for g to be some stronger forms of topological transitivity, sensitivity, ergodic, or mixing are obtained.