Thickly Syndetical Sensitivity of Topological Dynamical System

Abstract

Consider the surjective continuous map , where 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 is minimal and sensitive, then is syndetically sensitive. (2) Weak mixing implies thick sensitivity. (3) If is minimal and weakly mixing, then it is thickly syndetically sensitive. (4) If is a nonminimal -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.

1. 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 .

Throughout this paper, a topological dynamical system (TDS for short) is a pair where is a compact metric space without isolated points with metric and is a surjective continuous map. For any nonempty subset of and any , write and . When is a singleton, we write for .

Let be a subset of a positive integer set . is periodic if there is such that for some . is syndetic if it has bounded gaps; that is, there is such that for every . is thick if it contains arbitrarily long runs of positive integers; that is, there is a strictly increasing subsequence of such that . 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.

Write . A dynamical system is transitive if for each pair of nonempty open subsets of , is nonempty. A point is transitive if the orbit is dense in . A system is minimal if any is transitive. We say is mixing if for each pair of nonempty open subsets , is cofinite and is weakly mixing if is transitive. The set there exists an increasing sequencesuch that is said to be the -limit set of . We say is a minimal point (or almost periodic point), if the -limit set of is a minimal set.

Lemma 1. If is a transitive nonminimal TDS with as a transitive point and as a minimal set of , then is thick for every .

Proof. For any , since is uniformly continuous, so , such that for any , . For any transitive point and any minimal set , , such that , so ; that is, is thick.
Recall that the map induced by on is a nonempty compact subset} is defined by , . Then the pair is a dynamical system with the space endowed with the Hausdorff distance .

Lemma 2 (see [5]). is weakly mixing ⇔   is weakly mixing ⇔  is transitive.
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 .

In [6], Glasner and Weiss proved that nonminimal -system is sensitive. And various concepts of stronger sensitivity have been researched recently (see [5–15]).

Lemma 3. If is an -system, with as a minimal subset of and as a nonempty open subset of , then is thickly syndetic for any .

Proof. For any , since is uniformly continuous, so , such that for any , . For any transitive point and any minimal set , , such that , so there is a minimal point with . Since is almost periodic, there exists a syndetic set with , so . So we have that is thickly syndetic.

2. Stronger Versions of Sensitivity

Let be a dynamical system. Write . It is easy to see that is sensitive if and only if for some and every nonempty open set .

Definition 4. A TDS is(1)syndetically sensitive if is syndetic for some and every nonempty open set ,(2)thickly sensitive if is thick for some and every nonempty open set ,(3)thickly syndetically sensitive if is thickly syndetic for some and every nonempty open set ,(4)thickly periodically sensitive if is thickly periodic for some and every nonempty open set ,(5)cofinitely sensitive if is cofinite for some and every nonempty open set .
From the definition, mixing cofinite sensitivity thickly periodically sensitive thickly syndetical sensitivity thick sensitivity and syndetical sensitivity.

Theorem 5. If is minimal and sensitive, then is syndetically sensitive.

Proof. For sensitive constant , by sensitivity of , there exist and such that for every nonempty open set . So there exists an open subset such that . Because is minimal, there exists such that , so . Because is uniformly continuous, so such that for any , . Also, is a minimal point, so is syndetic. For any , we have , so ; that is, This means is syndetic.

Theorem 6. Weak mixing implies thick sensitivity.

Proof. Let and let   be a nonempty open set of . By Lemma 2, is transitive. is a fixed point of (). By density of transitive points of in , we have , a transitive point of . Because of Lemma 1, is thick; that is, is thick (because ).

Theorem 7. If is a minimal weakly mixing TDS, then it is thickly syndetically sensitive.

Proof. Since is a fixed point of , by the proof of Lemma 1, it is sufficient to prove that is syndetic for any open set of and any . Given and , let be a transitive point for . So there exist such that . Choose to be an modulus of uniform continuity for so that . Then for implies and . Since the orbit of is dense in , there are strictly increasing integers such that for . Hence, . Because is uniformly continuous, so such that for any , . Since is a minimal point, is syndetic. For any   (i.e., ), we have , that is, Then,

Theorem 8. If is a nonminimal -system, then it is thickly syndetically sensitive.

Proof. Let be minimal sets of with and let be a nonempty open set of . For any , since is uniformly continuous, so , such that for any , . By Lemma 3, we have , , thickly syndetic, so we have , syndetic. Thus, for every , . By arbitrary of , , is thickly syndetic; that is, is thickly syndetic.

Corollary 9. Devaney chaos (or -system without isolated points) implies thickly periodic sensitivity.

Proof. The proof is similar to Theorem 8.

3. 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 as a model for the circle . The metric is defined by . Rigid rotation by the real number is then given by Corresponding to the irrational , the Denjoy homeomorphism is an orientation preserving homeomorphism of the circle characterized by the properties: the rotation number of is ; there is a Cantor set on which acts minimally; and if and are any two components of then for some integer (see [4]). There is a Cantor function that semiconjugates with being a monotone surjection that collapses the components of (and so maps onto ) with .
Let be the minimal subsystem of a Denjoy homeomorphism and is a connected component. Then is syndetically sensitive but not thickly sensitive, and especially, is not thickly syndetically sensitive.

Proof. Let be a nonempty open set of . For any , there is such that . Let be the arc in whose endpoints are and and whose length is . Then there exist and such that . Let be one of the connected components of with diam and . is syndetic. For any , . So ; that is, is syndetically sensitive.
On the other hand, we assume that is thickly sensitive. Put . Choose an open set ( is little enough) of such that are pairwise disjoint where . Since is a rotation, are pairwise disjoint for every . So are pairwise disjoint for every . Because of thick transitivity, there exists an integer , such that , . Since are pairwise disjoint, so we have the length of arc . This is a contradiction.

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their helpful remarks. This work is supported by NSFC 11001038 and 11271061, the National High Technology Research and Development Program of China (863 Program) (2012AA01A309), and the Fundamental Research Funds for the Central Universities DC120101112.