Thickly Syndetical Sensitivity of Topological Dynamical System

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 nonminimalM-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.


Introduction and Preliminaries
The idea of sensitivity from the work [1,2] by Ruelle and Takens was applied to topological dynamics by Auslander and Yorke in [3] and popularized later by Devaney in [4].A system (, ) is called -sensitive if there exists a positive  (sensitive constant) such that any  ∈  is a limit of points  ∈  satisfying the condition (  (),   ()) >  for some positive integer .
Let  be a subset of a positive integer set Z + . is periodic if there is  such that { +  :  ∈ Z + } ⊂  for some  ∈ Z + . is syndetic if it has bounded gaps; that is, there is  such that {,  + 1, . . .,  + } ∩  ̸ = 0 for every  ∈ Z + . is thick if it contains arbitrarily long runs of positive integers; that is, there is a strictly increasing subsequence {  } of N such that  ⊃ ∪ ∞ =1 {  ,   + 1, . . .,   + }. is piecewise syndetic if it is an intersection of a syndetic set with a thick set. is thickly syndetic if for every  the positions where length  runs begin from a syndetic set. is thickly periodic if for every  the positions where length  runs begin from a periodic set.
Let (, ) be a transitive TDS, we say that the system is (1) a -system if the periodic points are dense in ; (2) an -system if the almost periodic points are dense in .

Lemma 3. If (𝑋, 𝑓
) is an -system, with  as a minimal subset of  and  as a nonempty open subset of , then (, (, )) is thickly syndetic for any  > 0.

Some Examples
Now we give an example which is syndetically sensitive but not thickly sensitive; especially, (  ,   ) is not thickly syndetically sensitive.
Example 10.We will use R/Z as a model for the circle  1 .The metric   is defined by   (, ) = min{|−|, 1−|−|}.Rigid rotation by the real number  is then given by Corresponding to the irrational , the Denjoy homeomorphism   :  1 →  1 is an orientation preserving homeomorphism of the characterized by the properties: the rotation number of   is ; there is a Cantor set   ⊂  1 on which   acts minimally; and if  and V are any two components of  1 \   then    () = V for some integer  (see [4]).There is a Cantor function ℎ  : S 1 →  1 that semiconjugates   with   : ℎ  being a monotone surjection that collapses the components of  1 \  (and so maps   onto  1 ) with   ∘ ℎ  = ℎ  ∘   .
Let (  ,   ) be the minimal subsystem of a Denjoy homeomorphism and  = max{diam() :  is a connected component of  Example 11.Every uniformly rigid weakly mixing minimal system (see [9], e.g., for the existence of these) is thickly syndetically sensitive but not cofinitely sensitive.
is easy to see that (, ) is sensitive if and only if (, ) ̸ = 0 for some  > 0 and every nonempty open set  ⊂ .(5)cofinitelysensitive if (, ) is cofinite for some  > 0and every nonempty open set  ⊂ .