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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 649348, 12 pages
http://dx.doi.org/10.1155/2010/649348
Research Article

Furstenberg Families and Sensitivity

1Department of Mathematics, Guangzhou University, Guangzhou 510006, China
2Department of Mathematics, South China Normal University, Guangzhou 526061, China

Received 31 August 2009; Revised 17 November 2009; Accepted 22 January 2010

Academic Editor: Yong Zhou

Copyright © 2010 Huoyun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system is -sensitive if there exists a positive such that for every and every open neighborhood of there exists such that the pair is not --asymptotic; that is, the time set belongs to , where is a Furstenberg family. A dynamical system is (, )-sensitive if there is a positive such that every is a limit of points such that the pair is -proximal but not --asymptotic; that is, the time set belongs to for any positive but the time set belongs to , where and are Furstenberg families.

1. Introduction

Throughout this paper a topological dynamical system (TDS) is a pair , where is a compact metric space with a metric and is a continuous surjective map. Let be the set of nonnegative integers.

The phrase—sensitive dependence on initial condition—was first used by Ruelle [1], to indicate some exponential rate of divergence of orbits of nearby points. Following the work by Guckenheimer [2], Auslander and Yorke [3], Devaney [4], a TDS is called sensitive if there exists a positive such that for every and every open neighborhood of , there exist and with ; that is, there exists a positive such that in any opene (= open and nonempty) set there are two distinct points whose trajectories are apart from (at least one moment).

Recently, several authors studied the sensitive property (cf. Abraham et al. [5], Akin and Kolyada [6]). The following proposition holds according to [6].

Proposition 1.1. Let be a TDS. The following conditions are equivalent.(1) is sensitive.(2)There exists a positive such that for every and every open neighborhood of there exists with .(3)There exists a positive such that in any opene set there exist and with .(4)There exists a positive such that in any opene set there exist with .

From Proposition 1.1, we know that a TDS is sensitive if and only if there exists a positive such that in any opene set there are two distinct points whose trajectories are infinitely many times apart at least of .

Some authors introduced concepts which link the Li-Yorke versions of chaos with the sensitivity in the recent years. Blanchard et al. [7] introduced the concept of spatiotemporal chaos. A TDS is called spatiotemporally chaotic if every is a limit of points such that the pair is proximal but not asymptotic; that is, the pair is a Li-Yorke scrambled pair [8]. That is

Akin and Kolyada [6] introduced the concept of Li-Yorke sensitivity. A TDS is called Li-Yorke sensitive if there is a positive such that every is a limit of points such that the pair is proximal but not -asymptotic. That is,

We see that Li-Yorke sensitivity clearly implies spatiotemporal chaos, but the latter property is strictly weaker (see [6]).

Let . The upper density of is

where denotes the cardinality of the set. The lower density of is

A pair is distributively scrambled pair [9] if there is positive such that , that is, , and for any positive .

Let be the set of nonnegative integers, and let be the collection of all subsets of . A subset of is called a Furstenberg family [10] if it is hereditary upwards; that is, and imply .

In the past few years, some authors [1114] investigated proximity, mixing, and chaos via Furstenberg family. In [13], -scrambled pair was defined via a Furstenberg family . A pair is called -scrambled pair if there is positive such that , and for any positive . In [14], -scrambled pair was defined via Furstenberg families and . A pair is called -scrambled pair if there is positive such that for any positive , and .

In this paper we investigate the sensitivity from the viewpoint of Furstenberg families.

A dynamical system is -sensitive if there exists a positive such that for every and every open neighborhood of there exists such that belongs to , where is a Furstenberg family.

A dynamical system is -sensitive if there is a positive such that every is a limit of points such that belongs to for any positive but belongs to , where and are Furstenberg families.

In Section 2, some basic notions related to Furstenberg families are introduced. In Section 3, we introduce and study the concept of -sensitivity. In Section 4, the notion of (, )-sensitivity is introduced and investigated, and the sensitivity of symbolic dynamics in the sense Furstenberg families is discussed finally.

2. Preliminary

In this section, we introduce some basic notions related Furstenberg families (for details see [10]). For a Furstenberg family , the dual family is

Clearly, if is a Furstenberg family then so is . Let be the collection of all subsets of . It is easy to see that . Clearly, and implies . Let be the family of all infinite subsets of . It is easy to see that is a Furstenberg family and is the family of all cofinite subsets.

A Furstenberg family is proper if it is a proper subset of . It is easy to see that a Furstenberg family is proper if and only if and . Any subset of can generate a Furstenberg family for some .

A Furstenberg family is countably generated [10, 13] if there exists a countable subset of such that . Clearly, is a countably generated proper family.

For Furstenberg families and , let . A Furstenberg family is full if it is proper and . It is easy to see that a Furstenberg family is full if and only if . Clearly, and are full. Clearly, if is full then . A Furstenberg family is a filterdual if is proper and .

For every , let

Clearly, and every is a full Furstenberg family (see [13]).

Let be a TDS and . We define the meeting time set

In particular we have for .

Let and . If , is called an -attaching point of . The set of all -attaching points of is called the set of -attaching of , denoted by . Clearly,

Let be a Furstenberg family. Recall that a TDS is -transitive if for each pair of opene subsets and of , . is -mixing if is -transitive.

Let be a TDS. A Furstenberg family is compatible with the system [13] if the set of -attaching of is a set of for each open set of .

3. -Sensitivity

In this section, we introduce and study the concept of -sensitivity. Let be a TDS and a Furstenberg family. Suppose that . Let . denotes the closure of . A subset of is called invariant for if .

We will use the following relations on :

For any subset and any point , we write

We define the sets of -asymptotic pairs

We say that is weakly -sensitive [10] if there is a positive —a weakly -sensitive constant—such that in every opene subset of there exist and of such that the pair is not --asymptotic. That is, , or .

We say that is -sensitive if there exists a positive —a -sensitive constant—such that for every and every open neighborhood of there exists such that the pair is not --asymptotic.

Theorem 3.1. Let be a TDS. Let and be Furstenberg families. Suppose that . If is weakly -sensitive, then that is -sensitive.

Proof. If is not -sensitive, then for each there exists a and there exists an open neighborhood of such that for each . Thus , this implies . Since , by the triangle inequality we have for any and of . Then . So , this contradicts the is weakly -sensitive.

Corollary 3.2. Let be a TDS and a filderdual. The system is weakly -sensitive if and only if it is -sensitive.

Lemma 3.3. Let be a TDS. A Furstenberg family is compatible with the system if and only if the set of -attaching of is an set of for each closed subset of .

Proof. Suppose that is a closed subset of , then
Hence, . Thus is a set of , if and only if is an set of .

Lemma 3.4. Let be a TDS and and Furstenberg families. Suppose that is compatible with the system , and . If is weakly -sensitive, then there exists a positive such that for every , is a dense set.

Proof. Since is compatible with the system , then is an set of by Lemma 3.3. Suppose that , where every is a closed subset of , then . Suppose that for each there exists such that is not first category. By Baire theorem there exists an opene subset of for some such that . Hence for each , . Since , by the triangle inequality we have for any and of . Then . So , this contradicts the that is weakly -sensitive.

The following lemma is proved in [13]. We give another proof here for completeness.

Lemma 3.5. Let be a TDS and a Furstenberg family. If is a countably generated proper family, or , then is compatible with the system .

Proof. (1) Let be a closed subset of . Suppose that is a proper family countably generated by , where is countable set, then

Hence, is an set.
(2) Suppose that . If , then . Since is a countably generated proper family, the result is true by (1).
Suppose that . It is easy to see that where Hence is an set.

By Lemma 3.3, 3.5 holds.

Example 3.6. Let and . If is weakly -sensitive, then (1) is -sensitive,(2)there exists a positive such that for every , is a dense set.
Since , then . Since , then . For any , then

Hence . If is weakly -sensitive, then is -sensitive by Theorem 3.1. By Lemmas 3.5 and 3.4 if is weakly -sensitive, then there exists a positive such that for every , is a dense set.

The following theorem is based on arguments in Huang and Ye [15]. It is called Huang-Ye equivalences in [6]. We state it here via Furstenberg families.

Theorem 3.7. Let be a TDS. If is a filterdual and is compatible with , then the following statements are equivalent.(1) is weakly -sensitive.(2)There exists a positive such that is a first category subset of .(3)There exists a positive such that for every ,  is a first category subset of .(4)There exists a positive such that for every , .(5)There exists a positive such that is dense in .(6) is -sensitive.

Proof. (1) (6). By Corollary 3.2, it holds.
(2) (1). If the system is not weakly -sensitive then for every , there exists an opene subset of such that for each , that is, . Then , this implies . Hence, . So is not of first category.
(3) (2). By Lemma 3.3, we know that is an set. Suppose that , where is a closed subset of . Then . If is not first category then by the Baire category theorem some has nonempty interior. If and , then . So is not first category.
(1) (3). By Lemma 3.4, it holds.
Thus, we have proved that (1)–(3) are equivalent.
(4) (1). If is not weakly -sensitive, then for any there exists an opene subset such that . Let . Then this implies .
(3) (4). If there exists a positive such that for every is a first category subset of , then is a dense subset of . Thus (4) is true.
(2) (5). At first, we note that
By Baire theorem, (2) (5).

Theorem 3.8. Let be a TDS. Suppose that have two nonempty invariant subsets and of with such that and are dense subsets of , then there exists a positive such that is a dense subset of , and if is a full Furstenberg family then is weakly -sensitive.

Proof. Since , there exist positive numbers and such that . Since and are dense subsets of , it is easy to check that so are and . Since , then is a dense subset of . Since is full then , this implies that is a dense subset of . Hence, is weakly -sensitive.

A map is semiopen if the image of an opene subset contains an opene subset. A factor map between dynamical systems is a continuous surjective map such that . The weakly -sensitivity can be lifted up by a semi-open factor map.

Theorem 3.9. Let and be TDS and semi-open factor map. Let be a Furstenberg family. If is weakly -sensitive, so is .

Proof. Let be a weakly -sensitive constant for . Since is continuous then there is such that if then .
Let be an opene subset of . As is semi-open, contains an opene subset of . Since is weakly -sensitive, then there exist and of such that . Let with and . Then , that is, is weakly -sensitive.

4. -Sensitivity

In this section, we introduce and study the notion of ()-sensitivity which links chaos and sensitivity via a couple Furstenberg families .

Let be a TDS and . A pair is called -proximal if

We denote the set of all -proximal pairs by .

The following lemma comes from [11].

Lemma 4.1. Let be a TDS and . Then
Let be a Furstenberg family. A pair is called -proximal if for any . We denote the set of all proximal pairs by .
Note that [12]:

Suppose that and are Furstenberg families.
A TDS is called -spatiotemporally chaotic if every is a limit of points such that the pair is -proximal but not -asymptotic. That is, for all

When , it is the usual spatiotemporal chaos.
A TDS is called -sensitive if there is a positive such that every is a limit of points such that the pair is -proximal but not --asymptotic.
That is, for all
When , is the usual Li-Yorke sensitivity.
If the pair is -proximal but not --asymptotic, then is the usual distributively scrambled pair.

We will use the following lemmas which comes from [10, 11], respectively.

Lemma 4.2. Let be a full Furstenberg family. If is -mixing, then is a dense set of for each and each .

Lemma 4.3. Let be a TDS and a Furstenberg family. is -transitive if and only if for every and every opene subset of , is an open and dense subset of (see [10, Proposition ]).

Theorem 4.4. Let be a TDS. Let and be Furstenberg families. If there exists a positive such that is a dense set for every , and is a dense set of for every , then is -sensitive.

Proof. Since is a dense subset of . Hence, is - sensitive.

Theorem 4.5. Let be a nontrivial TDS and a full filterdual. Suppose that is countably generated. If is -mixing, then is -sensitive.

Proof. Suppose that is a proper family countably generated by , where is a countable set. Then . By Lemmas 4.1 and 4.2, is a dense set of . Choose such that is a nonempty open subset of . By Lemma 4.3, is a dense subset of . By Theorem 3.7, is -sensitive. Thus is -sensitive by Theorem 4.4.

Lemma 4.6. Let be a TDS. Suppose that is a full Furstenberg family and is compatible with the system . If there is a fixed point of such that is dense subset of , then is a dense set of .

Proof. As is dense subset of , it is easy to check that so is for any positive . Since for some positive , then is a dense set of . As is full then , this implies that is a dense subset of . And since is compatible with the system , is a set of . By , then is a dense set of .

5. -Sensitivity of Symbolic Dynamics

Finally, as examples we will discuss the -sensitivity and ()-sensitivity of symbolic dynamics.

Let with the discrete topology. Let , for all . Let with the product topology. Then is a compact metric space. is called the symbolic space generated by . Let be the shift which will be defined as for any of . Then is called symbolic dynamics. Let .

We define a metric which is compatible with the product topology on as follows: for all

Theorem 5.1. Let be a full Furstenberg family. Then is -sensitive.

Proof. Let and . Then and are fixed points of , and both and are dense subsets of . By Theorem 3.8, is weakly -sensitive. Let be a weakly -sensitive constant. Now we show that is also -sensitive. For any of and for any open neighborhood of , there exist points and of such that . Choose of such that if otherwise . Then , so is -sensitive.

Lemma 5.2. Suppose that is a full Furstenberg family and is compatible with the system , then is a dense set of for every of .

Proof. By Lemma 4.6, is a set of for every of .

Now we show that is dense for every of . For any of and for any open neighborhood of . Choose of such that , then for any positive , this implies that is dense. Since is a full then , so is dense set of . By , then is a dense set of .

Lemma 5.3. Suppose that is a full Furstenberg family and is compatible with the system , then there exists a positive such that for every , is a dense set of .

Proof. Let and . Then and are fixed points of , and both and are dense subsets of . By Theorem 3.8 there exists a positive such that is a dense set of . Now we show for every , is a dense set of . For any of , and for any open neighborhood of of , there exists of such that . Choose of such that, if otherwise , when . Then . Since is full, then , this implies that . Hence is a dense set of . Since is compatible with the system , then is a set of . Hence is a dense set of .

By Lemmas 5.2 and 5.3, the following theorem holds.

Theorem 5.4. Suppose that and are full, and are compatible with , then is -sensitive. In particular, is -sensitive.

Acknowledgments

The authors greatly thank the referees for the careful reading and many helpful remarks. This work was supported by the National Nature Science Funds of China (10771079, 10471049), and Guangzhou Education Bureau (08C016).

References

  1. D. Ruelle, “Dynamical systems with turbulent behavior,” in Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), vol. 80 of Lecture Notes in Physics, pp. 341–360, Springer, Berlin, Germany, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Guckenheimer, “Sensitive dependence to initial conditions for one-dimensional maps,” Communications in Mathematical Physics, vol. 70, no. 2, pp. 133–160, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Auslander and J. A. Yorke, “Interval maps, factors of maps, and chaos,” The Tôhoku Mathematical Journal, vol. 32, no. 2, pp. 177–188, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Devaney, Chaotic Dynamical Systems, Addison-Wesley, Redwood City, Calif, USA, 1980.
  5. C. Abraham, G. Biau, and B. Cadre, “On Lyapunov exponent and sensitivity,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 395–404, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. Akin and S. Kolyada, “Li-Yorke sensitivity,” Nonlinearity, vol. 16, no. 4, pp. 1421–1433, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. F. Blanchard, E. Glasner, S. Kolyada, and A. Maass, “On Li-Yorke pairs,” Journal für die Reine und Angewandte Mathematik, vol. 547, pp. 51–68, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B. Schweizer and J. Smítal, “Measures of chaos and a spectral decomposition of dynamical systems on the interval,” Transactions of the American Mathematical Society, vol. 344, no. 2, pp. 737–754, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, The University Series in Mathematics, Plenum Press, New York, NY, USA, 1997. View at MathSciNet
  11. W. Huang, S. Shao, and X. Ye, “Mixing and proximal cells along sequences,” Nonlinearity, vol. 17, no. 4, pp. 1245–1260, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Shao, “Proximity and distality via Furstenberg families,” Topology and Its Applications, vol. 153, no. 12, pp. 2055–2072, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. C. Xiong, J. Lü, and F. Tan, “Furstenberg familes and chaos,” Science in China Series A, vol. 50, pp. 1352–1333, 2007. View at Google Scholar
  14. F. Tan and J. Xiong, “Chaos via Furstenberg family couple,” Topology and Its Applications, vol. 156, no. 3, pp. 525–532, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W. Huang and X. Ye, “Devaney's chaos or 2-scattering implies Li-Yorke's chaos,” Topology and Its Applications, vol. 117, no. 3, pp. 259–272, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet