About this Journal Submit a Manuscript Table of Contents
ISRN Applied Mathematics
Volume 2013 (2013), Article ID 650197, 4 pages
http://dx.doi.org/10.1155/2013/650197
Research Article

On the Attractor of the Product System

1Department of Basic Science, Dalian Naval Academy, Dalian, Liaoning 116018, China
2Department of Mathematics, Dalian Nationalities University, Dalian, Liaoning 116600, China

Received 1 June 2013; Accepted 7 August 2013

Academic Editors: G. Stavroulakis and W. Yeih

Copyright © 2013 Guifeng Huang and Lidong Wang. 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 investigate the attractor of the product system and mainly prove that the likely limit set of the product system equals the product of one of each factor system for compact systems with solenoid attractors. Specially, this holds for the product map of Feigenbaum maps. Furthermore, we deduce that the Hausdorff dimension of the likely limit set of the product map for Feigenbaum maps is the sum of one of each factor map.

1. Introduction

An attractor plays a very important role in the studies of dynamical system. The concept of the likely limit set was introduced by Milnor in 1985 (see [1]). As indicated in [1], the likely limit set always exists. Because this kind of attractors gathers the asymptotic behaviors of almost all points, it is very necessary to study them.

In [2], an analytic function similar to a unimodal Feigenbaum map (we call the solution of Feigenbaum functional equation Feigenbaum map) was investigated; the Hausdorff dimension of the likely limit set of it was estimated. We discussed the dynamical properties of unimodal Feigenbaum map, estimated the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum map, and proved that for every , there always exists a unimodal Feigenbaum map such that the Hausdorff dimension of the likely limit set is . We also considered the kneading sequences of unimodal Feigenbaum maps (see [3]). Similarly to this, we studied the nonunimodal Feigenbaum maps in [4].

In this paper, we explore the dynamics of the product map whose every factor map has solenoid attractor and show that, for this kind of product map, the likely limit set has multiplicative property; that is, the likely limit set of the product map equals the product of one of each factor system. As an application, we consider the product of Feigenbaum maps. The main results are Theorems 8 and 10.

2. Basic Definitions and Preparations

Milnor introduced the concept of the likely limit set in 1985 [1].

Definition 1. Let be a compact manifold (with boundary possibly) and a continuous map of into itself. The likely limit set of is the smallest closed subset of with the property that for every point outside of a set of Lebesgue measure zero ( denotes the -limit set of the point under ).

A set is called a minimal  set of if and for every point . As is well known, the minimal set is a nonempty, closed, and invariant subset under , and it has no proper subset with these three properties (see [5]). Therefore, if is a minimal set with for almost all , then .

The notion of period interval is an extension of one of period point [6].

Definition 2. Let be a real compact interval and the set of continuous maps from into itself. Let be the set of positive integers, . A sequence of closed subintervals of is said to be periodic of if they have disjoint interiors and for any and . Call a cycle of periodic intervals.
is called a solenoid of if there are a strictly increasing sequence of positive integers and periodic sequences of closed intervals of period such that for any and .
We call the family a covering of of type . If a solenoid admits a covering of type , we call it a doubling period solenoid of .

Let , be , dimensional compact space, respectively. Descartes product of and is ; denotes Descartes product of and .

Let , be continuous maps. For all ,

The concepts of -limit set and likely limit set in one dimension can be extended to the product space.

In 1978, Feigenbaum [7] put forward Feigenbaum functional equation: where is to be determined.

In 1985, Yang and Zhang [8] proposed the second type of Feigenbaum functional equation: where is to be determined.

There is a close link between the solutions of these two types of equations.

Lemma 3. Let be a unimodal Feigenbaum map. If, for , , and, for , (considering the left or right derivative at the end points), then(1)the likely limit set is a minimal set of ;(2),
where and denotes the Hausdorff dimension.

For a proof see [3].

Lemma 4. For every , there exists a unimodal Feigenbaum map such that .

For a proof see [3].

Lemma 5. Let be a nonunimodal Feigenbaum map and the minimum point of on . If, for , and, for , (considering the left or right derivative at the end points), then(1)there exists a set of contractions such that its invariant set is the likely limit set and a minimal set of ;(2),
where and denotes the Hausdorff dimension.

For a proof see [4].

Lemma 6. For every , there always exists a nonunimodal Feigenbaum map such that .

For a proof see [4].

Lemma 7. If , are well-distributed Cantor sets, then where denotes the Hausdorff dimension.

For a proof see [9].

3. The Theorems and Their Proofs

Theorem 8 (main theorem). Let , be compact intervals, , and let , be continuous, and let , be solenoid attractors; then for , one has

Proof. Let , be as indicated in the theorem.
Consider where , are strictly increasing sequences of positive integer; , are two sequences of closed intervals with period and , respectively.
First, we show .
By the definition of likely limit set, we assume that for every outside of the set with Lebesgue measure zero and for every outside of the set with Lebesgue measure zero. Since and are closed sets, is a compact invariant set; then it is a closed set. is the smallest invariant closed set that attracts almost all points in . Thus, it is enough to prove that attracts almost all points in in order to show .
Because any , , any , , for any , it is easy to know Since is a Lebesgue measure zero set, it follows that attracts almost all points in . Therefore,
Second, we prove that .
Suppose that there is a point , but . is a closed set, so there exists an open neighborhood such that and .
By the definition of solenoid attractor, there must be period intervals , containing , , respectively. Assume that the period of is , the one of is ; then the one of is not more than , where . So for any , has positive Lebesgue measure, this shows the neighborhood of attracts a positive measure set, contrary to the definition of . Therefore, this assumption does not hold. So
Sum up, we obtain

Obviously, this conclusion holds for finite maps.

Corollary 9. Let be a continuous map, where is a compact interval and is a solenoid attractor, ; then

Specially, this holds for the unimodal Feigenbaum map and the nonunimodal Feigenbaum map which we have discussed in [3, 4], respectively.

Theorem 10. Let or ; let , be the unimodal Feigenbaum map or the nonunimodal Feigenbaum map as in Lemma 3 or Lemma 5, ; then

Proof. Let . First, we show is a doubling period solenoid of if is a unimodal Feigenbaum map of into itself. Let .
By the proof of Lemma 3 (see [3]), we know For any ,  , there exists such that ; then
It is easy to see that and is the sum of closed sets which are disjoint from each other; that is, is a sequence of closed intervals with period . is obtained by removing an open interval in each connected component of and . So is a doubling period solenoid of .
Similarly to this, we can also show that is a doubling period solenoid of if is a nonunimodal Feigenbaum map of into itself.
Hence, by Theorem 8 we get

Theorem 11. If , are the unimodal Feigenbaum maps as in Lemma 3 or the nonunimodal Feigenbaum maps as in Lemma 5, then

Proof. It is easy to know from Theorem 10 and Lemma 7.

By Lemmas 4 and 6 and Theorem 11, we immediately have the following.

Corollary 12. For any , there are Feigenbaum maps , such that

4. Conclusion

We discussed the likely limit set of product system, show that the likely limit set of product map of two maps with solenoid likely limit sets equals the product of one of each factor map, apply this to the unimodal Feigenbaum map and the nonunimodal Feigenbaum map, and know that the Hausdorff dimension of the likely limit set of the product map of two Feigenbaum maps is the sum of one of each factor map.

Acknowledgment

The authors wish to thank the NSFC (no. 11271061) for financial support.

References

  1. J. Milnor, “On the concept of attractor,” Communications in Mathematical Physics, vol. 99, no. 2, pp. 177–195, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. K. J. Falconer, The Geometry of Fractal Sets, vol. 85, Cambridge University Press, Cambridge, UK, 1986. View at MathSciNet
  3. G. F. Huang, L. D. Wang, and G. F. Liao, “A note on the unimodal Feigenbum's maps,” International Journal of Modern Physics B, vol. 14, pp. 3101–3111, 2009.
  4. G. F. Huang and L. D. Wang, “On the attractors of Feigenbaum maps,” Annales Polonici Mathematici. In press.
  5. L. S. Block and W. A. Coppel, Dynamics in One Dimension, vol. 1513 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1992. View at MathSciNet
  6. V. J. López and L. Snoha, “There are no piecewise linear maps of type 2,” Transactions of the American Mathematical Society, vol. 349, no. 4, pp. 1377–1387, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” Journal of Statistical Physics, vol. 19, no. 1, pp. 25–52, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. Yang and J. Z. Zhang, “The second type of Feigenbaum's functional equations,” Science China, vol. 12, pp. 1061–1069, 1985 (Chinese).
  9. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, UK, 1990. View at MathSciNet