Thickly Syndetical Sensitivity of Topological Dynamical System

Abstract EN

Consider the surjective continuous map f:X→X, where X is a compact metric space.

In this paper we give several stronger versions of sensitivity, such as thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, and strong sensitivity.

We establish the following.

(1) If (X,f) is minimal and sensitive, then (X,f) is syndetically sensitive.

(2) Weak mixing implies thick sensitivity.

(3) If (X,f) is minimal and weakly mixing, then it is thickly syndetically sensitive.

(4) If (X,f) is a nonminimal M-system, then it is thickly syndetically sensitive.

Devaney chaos implies thickly periodic sensitivity.

(5) We give a syndetically sensitive system which is not thickly sensitive.

(6) We give thickly syndetically sensitive examples but not cofinitely sensitive ones.