International Journal of AnalysisVolume 2014 (2014), Article ID 538691, 6 pageshttp://dx.doi.org/10.1155/2014/538691

Research Article

On Nonautonomous Discrete Dynamical Systems

Dhaval Thakkar¹ and Ruchi Das²

¹Vadodara Institute of Engineering, Kotambi, Vadodara 391510, India²Department of Mathematics, Faculty of Science, The M.S. University of Baroda, Vadodara 390002, India

Received 26 November 2013; Accepted 18 April 2014; Published 2 June 2014

Academic Editor: Harumi Hattori

Copyright © 2014 Dhaval Thakkar and Ruchi Das. 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 define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space.

1. Introduction

In the recent past, lots of studies have been done regarding dynamical properties in nonautonomous discrete dynamical systems. In [1], Kolyada and Snoha gave definition of topological entropy in nonautonomous discrete systems. In [2], Kolyada et al. discussed minimality of nonautonomous dynamical systems. In [3, 4], authors studied -limit sets in nonautonomous discrete systems, respectively. In [5], Krabs discussed stability and controllability in nonautonomous discrete systems. In [6, 7], Huang et al. studied topological pressure and preimage entropy of nonautonomous discrete systems. In [8–15], authors studied chaos in nonautonomous discrete systems.

In [16], Liu and Chen studied -limit sets and attraction of nonautonomous discrete dynamical systems. In [8, 17] authors studied weak mixing and chaos in nonautonomous discrete systems. In [18] Yokoi studied recurrence properties of a class of nonautonomous discrete systems. Recently in [19] we defined and studied expansiveness, shadowing, and topological stability in nonautonomous discrete dynamical systems given by a sequence of continuous maps on a metric space.

In this paper we define and study expansiveness, shadowing, and topological stability in nonautonomous discrete dynamical systems given by a sequence of homeomorphisms on a metric space. In the next section, we define and study expansiveness of a time varying homeomorphism on a metric space. In section following to the next section, we define and study shadowing or P.O.T.P. for a time varying homeomorphism on a metric space. In the final section, we study topological stability of a time varying homeomorphism on a compact metric space.

2. Expansiveness of a Nonautonomous Discrete System Induced by a Sequence of Homeomorphisms

Throughout this paper we consider to be a metric space and to be a sequence of homeomorphisms, , where we always consider to be the identity map on and to be a time varying homeomorphism on . We denote For any , we define and, for , we define to be the identity map on . For time varying homeomorphism on , its inverse map is given by . For any , we define a time varying map (th-iterate of ) on , where Thus , for , and, for , . Also, for , , where each is the identity map on .

Definition 1. Let be a metric space and a sequence of maps, . For a point, let then the sequence , denoted by , is said to be the orbit of under time varying homeomorphism .

Definition 2. Let be a metric space and a sequence of homeomorphisms, . The time varying homeomorphism is said to be expansive if there exists a constant (called an expansive constant) such that, for any , , for some . Equivalently, if, for , for all then .

Remark 3. If in the above definition , for all , where is homeomorphism, then expansiveness of time varying homeomorphism on is equivalent to expansiveness of on [20].

Remark 4. Note that expansiveness of a time varying homeomorphism is independent of the choice of metric for if is compact.

Definition 5. Let and be two metric spaces. Let and be time varying homeomorphisms on and , respectively. If there exists a homeomorphism such that for all then and are said to be conjugate with respect to the map or -conjugate. In particular, if is a uniform homeomorphism, then and are said to be uniformly conjugate or uniformly -conjugate. (Recall that homeomorphism , such that and are uniformly continuous, is called a uniform homeomorphism.)

For example, if on and on are time varying homeomorphisms, then is uniformly -conjugate to , where is defined by .

Theorem 6. Let and be metric spaces. Let and be time varying homeomorphisms on and , respectively, such that is uniformly conjugate to . Then is expansive on if and only if is expansive on .

Proof. Since is uniformly conjugate to , therefore there exists a uniform homeomorphism sach that , for all , which implies that , for all , and , for all . Now, for all , and similarly, for all , we also have So we get , for all . Similarly, , for all . Suppose is expansive on with expansive constant . Since is uniformly continuous, therefore there exists a such that, for any with , . Let such that ; then and since is expansive on ; there exists such that which implies that . Hence is expansive on .Conversely, suppose is expansive on with expansive constant . Since is uniformly continuous, there exists such that, for any with , . For any with , observing that , it follows that there exists such that which implies that . Thus is expansive on .

Corollary 7. Let be a compact metric space, a metric space, a time varying homeomorphism on , and a homeomorphism. If is expansive on , then is expansive on .

Theorem 8. Let be a compact metric space and a family of self-homeomorphisms on . Then time varying map is expansive if and only if is expansive.

Proof. Let be an expansive constant for . It is easy to verify that , for all . Let , ; then there is some such that ; that is, , for some , which implies is also expansive.Using the above Theorem, analogous to Theorem 2.2 in [19], we have the following result.

Theorem 9. Let be a compact metric space, an equicontinuous family of self-maps on , and an integer. Then time varying homeomorphism is expansive if and only if is expansive for any .

Definition 10. Let be a metric space, a time varying homeomorphism on , and a subset of . Then is said to be invariant under if (and therefore ), for all , and equivalently , for all .

We have the following result from the definition of invariance.

Theorem 11. Let be a metric space, a time varying homeomorphism which is expansive on , and an invariant subset of ; then restriction of to , defined by , is expansive.

Similar to Theorem 2.4 in [19], we have the following result.

Theorem 12. Let and be metric spaces and , time varying homeomorphisms on and , respectively. Consider the metric on defined by Then the time varying homeomorphism is expansive on if and only if and are expansive on and , respectively. Hence every finite direct product of expansive time varying homeomorphisms is expansive.

We have the following result for time varying homeomorphism similar to that for expansive homeomorphism on compact metric space [21].

Theorem 13. Let be a compact metric space and a time varying homeomorphism which is expansive on . If is the least upper bound of the expansive constants for , then is not an expansive constant for .

Proof. Let be an expansive constant for and ; then is also an expansive constant for . Let , for . Since is not an expansive constant for , therefore for each there exist such that for each integer . Also, for each , there exists an integer such that . Let and . We can assume that the sequences and converge to and , respectively, and we note that . Let be an arbitrary integer and an arbitrary positive number. Choose , , and with the following properties.   ,    implies , and    implies and implies . Let ; then Thus and therefore is not an expansive constant for .The topological analogue of generator was defined and studied by Keynes and Robertson [22]. We define and study this notion for invertible nonautonomous discrete dynamical system.

Definition 14. Let be a compact metric space and a time varying homeomorphism on . A finite open cover of is said to be a generator for if, for every bisequence of members of , is at most one point, where denotes the closure of .

Definition 15. Let be a compact metric space and a time varying homeomorphism on . A finite open cover of is said to be a weak generator for if for every bisequence of members of , is at most one point.

Theorem 16. Let be a compact metric space and a time varying homeomorphism on . Then the following are equivalent. (1) is expansive.(2) has a generator.(3) has a weak generator.

Proof. follows by definitions of generator and weak generator. We prove that . Let be a weak generator for and a Lebesgue number for . Let be a finite open cover by sets with . If is a bisequence of members of , then for every there is such that , and so . Since contains almost one point, therefore also contains atmost one point and hence is a generator.Next we prove that . Let be an expansive constant for and a finite open cover of by open balls of radius . Suppose , where ; then for every and since is expansive with expansive constant , we have .: suppose is a weak generator. Let be a Lebesgue number for . If , for all , then for every there is such that and so which is at most one point implying .

Example 17. Let , where , such that , for and , be a time varying homeomorphism on . Now note that and for all . Let be a finite open cover of with Lebesgue’s number . Note that and uniformly on . Then there exists such that implies and , for any . Since is Lebesgue’s number of , there are and in such that and . Now since is uniformly equicontinuous family, there exists such that implies , for any . Let , , such that . Then, for any , there exists such that . Thus , . Now put Now note that Thus cannot be a weak generator for . Therefore has no weak generator and hence by the above result is a nonexpansive time varying homeomorphism.

3. Pseudo Orbit Tracing Property of a Nonautonomous Discrete System by a Sequence of Homeomorphisms (P.O.T.P.)

Definition 18. Let be a metric space and a time varying homeomorphism on . For , the sequence in is said to be a -pseudo orbit of if For given , a -pseudo orbit is said to be -traced by if for all .The time varying homeomorphism is said to have shadowing property or pseudo orbit tracing property (P.O.T.P.) if, for every , there exists a such that every -pseudo orbit is -traced by some point of .

Remark 19. Note that in is a -pseudo orbit of if for we have and for we have .

Remark 20. If in the above definition , for all , where is homeomorphism, then P.O.T.P. of time varying homeomorphism on is equivalent to P.O.T.P. of on [20].

Remark 21. Note that shadowing property of time varying homeomorphism is independent of choice of metric if is compact.

Theorem 22. Let be a time varying homeomorphism on a metric space uniformly conjugate to time varying homeomorphism on metric space . Then has shadowing property if and only if has shadowing property.

Proof. Given any , applying the uniform continuity of implies that there exists such that, for any with , . As has P.O.T.P., there exists such that every -pseudo orbit of is -traced by some point of . Noting the fact that is uniformly continuous, there exists such that, for any with , .Now, we assert that every -pseudo orbit of is -traced by some point of .In fact, for any -pseudo orbit of , applying it follows that is a -pseudo orbit of . Then there exists such that is -traced by . This implies that, for any , , .By similar arguments using uniform continuity of one can prove the converse.Let and be metric spaces. Define metric on by By similar arguments given in Theorem 3.2 in [19] we can prove the following result.

Theorem 23. Let , be time varying homeomorphisms. Then and have shadowing property if and only if the time varying homeomorphism has shadowing property on . Hence every finite direct product of time varying homeomorphisms having shadowing property, has shadowing property.

Theorem 24. Let be the time varying homeomorphism on a metric space . Then has P.O.T.P. if and only if has P.O.T.P.

Proof. The proof follows observing that is a -pseudo orbit of if and only if is a -pseudo orbit of .Using the above result and similar arguments given in Theorem 3.3 in [19] we get the following result.

Theorem 25. Let be a compact metric space and an equicontinuous family of self-homeomorphisms on . Then time varying homeomorphism has P.O.T.P. if and only if has P.O.T.P. for any .

4. Topological Stability of Nonautonomous Discrete System Induced by a Sequence of Homeomorphisms on a Compact Metric Space

Let be a compact metric space, and define standard bounded metric on by And let be the space of all homeomorphisms on , where metric is defined by Let be the collection of all time varying homeomorphisms on . We define a metric on as follows:

For and ,

Definition 26. A time varying homeomorphism is said to be topologically stable in if for every there exists a , , such that for a time varying homeomorphism with there is a continuous map so that, for all , and , for all .

Theorem 27. Let be a compact metric space and let be a time varying homeomorphism on which is expansive and has shadowing then is topologically stable in .

Proof. Let be an expansive constant for time varying homeomorphism . choose such that . Since has shadowing property therefore for above we get , be a real number such that every -pseudo orbit of can be -traced by some orbit. Using expansiveness of ,one can prove that -pseudo orbit is -traced by unique .Let be a time varying map on such that . Let Since therefore is a -pseudo orbit for .Let be unique element of whose -orbit -traces . So we get a map with , for and . Letting , we have , for each .By similar arguments given in Theorem 4.1 in [19] we can prove that is continuous.

Conflict of Interests

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

Acknowledgments

The second author is supported by UGC Major Research Project F.N. 42–25/2013 for carrying out this research. The authors thank one of the referees for some useful suggestions.

References

S. Kolyada and L. Snoha, “Topological entropy of nonautonomous dynamical systems,” Random & Computational Dynamics, vol. 4, no. 2-3, pp. 205–233, 1996.

S. Kolyada, L. Snoha, and S. Trofimchuk, “On minimality of nonautonomous dynamical systems,” Nonlinear Oscillations, vol. 7, no. 1, pp. 86–92, 2004.

R. Kempf, “On -limit sets of discrete-time dynamical systems,” Journal of Difference Equations and Applications, vol. 8, no. 12, pp. 1121–1131, 2002.

J. S. Cánovas, “On -limit sets of non-autonomous discrete systems,” Journal of Difference Equations and Applications, vol. 12, no. 1, pp. 95–100, 2006.

W. Krabs, “Stability and controllability in non-autonomous time-discrete dynamical systems,” Journal of Difference Equations and Applications, vol. 8, no. 12, pp. 1107–1118, 2002.

X. Huang, X. Wen, and F. Zeng, “Topological pressure of nonautonomous dynamical systems,” Nonlinear Dynamics and Systems Theory, vol. 8, no. 1, pp. 43–48, 2008.

X. Huang, X. Wen, and F. Zeng, “Pre-image entropy of nonautonomous dynamical systems,” Journal of Systems Science & Complexity, vol. 21, no. 3, pp. 441–445, 2008.

F. Balibrea and P. Oprocha, “Weak mixing and chaos in nonautonomous discrete systems,” Applied Mathematics Letters, vol. 25, no. 8, pp. 1135–1141, 2012.

J. S. Cánovas, “Li-Yorke chaos in a class of nonautonomous discrete systems,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 479–486, 2011.

J. S. Cánovas, “Recent results on non-autonomous discrete systems,” Boletín de la Sociedad Española de Matemática Aplicada, no. 51, pp. 33–40, 2010.

J. Dvořáková, “Chaos in nonautonomous discrete dynamical systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4649–4652, 2012.

P. Oprocha and P. Wilczyński, “Chaos in nonautonomous dynamical systems,” Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, vol. 17, no. 3, pp. 209–221, 2009.

Y. Shi and G. Chen, “Chaos of time-varying discrete dynamical systems,” Journal of Difference Equations and Applications, vol. 15, no. 5, pp. 429–449, 2009.

C. Tian and G. Chen, “Chaos of a sequence of maps in a metric space,” Chaos, Solitons and Fractals, vol. 28, no. 4, pp. 1067–1075, 2006.

X. Wu and P. Zhu, “Chaos in a class of non-autonomous discrete systems,” Applied Mathematics Letters, vol. 26, no. 4, pp. 431–436, 2013.

L. Liu and B. Chen, “On -limit sets and attraction of non-autonomous discrete dynamical systems,” Journal of the Korean Mathematical Society, vol. 49, no. 4, pp. 703–713, 2012.

M. Murillo-Arcila and A. Peris, “Mixing properties for nonautonomous linear dynamics and invariant sets,” Applied Mathematics Letters, vol. 26, no. 2, pp. 215–218, 2013.

K. Yokoi, “Recurrence properties of a class of nonautonomous discrete systems,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 20, no. 4, pp. 689–705, 2013.

D. Thakkar and R. Das, “Topological stability of a sequence of maps on a compact metric space,” Bulletin of Mathematical Sciences, vol. 4, no. 1, pp. 99–111, 2014.

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North-Holland, Amsterdam, The Netherlands, 1994.

B. F. Bryant, “Expansive self-homeomorphisms of a compact metric space,” The American Mathematical Monthly, vol. 69, no. 5, pp. 386–391, 1962.

H. B. Keynes and J. B. Robertson, “Generators for topological entropy and expansiveness,” Mathematical Systems Theory, vol. 3, pp. 51–59, 1969.