Sensitivity for topologically double ergodic dynamical systems

As a stronger form of multi-sensitivity, the notion of ergodic multi-sensitivity (resp. strongly ergodically multi-sensitivity) is introduced. In particularly, it is proved that every topologically double ergodic continuous selfmap (resp. topologically double strongly ergodic selfmap) on a compact metric space is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive). And for any given integer m >= 2, f is ergodically multi-sensitive (resp. strongly ergodically multi-sensitive) if and only if so is f(m). Also, it is shown that if f is a continuous surjection, then f is ergodically multisensitive (resp. strongly ergodically multi-sensitive) if and only if so is sigma(f), where sigma(f) is the shift selfmap on the inverse limit space (lim) under left arrow (X, f). Moreover, it is proved that if f : X -> X (resp. g : Y -> Y) is a map on a nontrivial metric space (X, d) (resp. (Y, d')), and pi is a semiopen factor map between (X, f) and (Y, g), then the ergodic multi-sensitivity (resp. the strongly ergodic multi-sensitivity) of g implies the same property of f.

Automated summary

In this paper, the concepts of ergodic multi-sensitivity and strongly ergodically multi-sensitivity are introduced, with special cases proven in compact metric spaces. It is also demonstrated how the properties of different spaces and maps are related to each other.