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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 673578, 7 pages
Research Article

DC3 Pairs and the Set of Discontinuities in Distribution Functions

1Department of Mathematics, Dalian Nationalities University, Dalian 116600, China
2AGH University of Science and Technology, Faculty of Applied Mathematics, Aleja A. Mickiewicza 30, 30-059 Kraków, Poland
3Institute of Mathematics, Polish Academy of Sciences, Ulica Śniadeckich 8, 00-956 Warszawa, Poland
4School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 18 April 2013; Accepted 21 June 2013

Academic Editor: Guang Zhang

Copyright © 2013 Lidong 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.


We study the relationship between DC3 pairs and the set of discontinuities in distribution function. We also check relations between DC3 pairs for a continuous map and its higher iterates.

1. Introduction

In 1994, Schweizer and Smítal extended the definition of Li-Yorke pair for interval maps [1]. The main motivation was that chaotic dynamics, as introduced by Li and Yorke in [2], may be present in an interval map with zero topological entropy, while the adjusted definition can appear only in an interval map with positive topological entropy. The case of interval maps is very special, since in this context there is no difference between maps with DC1 pairs (the strongest possibility of distributional chaos) and DC3 pairs (the weakest possibility).

Let us first give the concepts that originated from [1] (but using modern terminology), since they are the main topics of the present paper.

Suppose that is a dynamical system, that is, a continuous map acting on a compact metric space (basic definitions related to dynamical systems, such as orbit and -limit set, can be found in any standard book on dynamical systems, e.g., [3]). For any positive integer , points , and real number , let where as usual denotes the cardinality of a set . If the map is clear from the context, we simply write and .

Definition 1. If a pair of points fulfills one of the following conditions: (DC1) for all and for some , (DC2) for all and for some , (DC3) for all , where is some nondegenerate interval, then we say that is a DC1, DC2, or DC3 pair, respectively.

In recent years many authors were interested in systems with DC pairs. While there are numerous results on properties of DC1 and DC2 pairs, not many are known about systems with only DC3 pairs. The reason is that if a DC3 pair can be detected, then there usually also exist DC2 pairs in the system.

By the definition we immediately have that DC1 implies DC2 and DC2 implies DC3, and it is also known that none of the reverse implications holds (e.g., see [4]). It can also be proved that DC1 or DC2 implies chaos in the sense of Li and Yorke, but DC3 does not [5]. Furthermore, recent result of Downarowicz shows that positive topological entropy is a sufficient condition for large set of DC2 pairs (so-called scrambled set of type ) in the system [6]. In [7, 8] the author shows that strong mixing properties, for example, specification property or topological exactness, are sufficient for scrambled sets of type . In [5] there is an example of distal map with DC3 pairs; hence, DC3 does not imply positive topological entropy or Li-Yorke pairs. Moreover, DC1 need not imply positive topological entropy, even in minimal systems (e.g., see [9]).

In this paper, we investigate the relationship between DC3 pairs and the set of discontinuities in distribution function. This will highlight many problems which can arise when one looks for numerical evidence of distributional chaos.

2. Distributional Chaos of Type 3

In this section we will focus on properties of distribution functions and , which may cause many problems during numerical investigation of the dynamics.

2.1. Discontinuity Points

The essential ingredient of all the three definitions of DC pairs is “sufficiently large” difference in values of functions and . Accordingly, an important question is how much values of and can differ if is not a DC3 pair. The following observation provides an upper bound.

Proposition 2 (see Lemma 1 of [10]). The following conditions are equivalent:(1)  is not a DC3 pair; (2)the set is at most countable.

Proof. Implication (2) (1) is trivial.
Conversely, assume that is not DC3 pair. Let , be the set of discontinuity points of and , respectively. If , then , because otherwise they would be different on an open interval containing . Since both functions are nondecreasing, they can have at most countably many discontinuity points, which ends the proof.

As we see, that for some values of parameter need not be enough for the occurrence of DC3 pair. If we try to predict distributional chaos numerically, then the parameter value we consider may be a discontinuity point of function or , and the pair is not DC3. Then we may think that the system has DC3 pairs while it does not. Therefore, to ensure ourselves that considered pair is DC3, we can pick another parameter value and repeat simulation. But again it can be another discontinuity point, and so on. From one point of view the set of such discontinuity points is small (it has Lebesgue measure zero), so in perfect situation the chance of picking up such a point is zero. However, if the set of discontinuity points may coincide with, say, for some , then all the points in that can be considered for computer simulation are wrong. So the first question is whether there really is a risk of such situation.

For any set , we denote its characteristic function by .

Theorem 3. For any there is a map without DC3 pairs and two points such that and .

Proof. Fix any increasing sequence such that . Let be a decreasing sequence such that for all . Now let and where Let be the connect-the-dots map defined by the following points; that is, is linear on countably many intervals with values at the endpoints of these intervals given by Then we see that for any its -limit set is the singleton consisting of one of the points . Similarly we can verify that and so has zero topological entropy. Then it has no DC3 pair by [1].
Consider the pair . Then .
We can see that Therefore we can easily verify that Thus and . On the other hand, for any we have that , provided that is sufficiently large. This gives for and for .

We can extend the construction in Theorem 3 to the following.

Theorem 4. For any finite sequence , there is a map without DC3 pairs and points such that for and for all other parameter values .

Proof. We only sketch the idea of this construction, leaving exact calculations to the reader.
First, extend the sequence to have elements for some . Let be a piecewise linear map of type in the Sharkovsky ordering (see [3]). Then we know that this map has topological entropy zero (thus has no DC3 pair) and a cycle consisting of exactly elements contained in the interval . We may assume that are fixed points of . We can also transform the interval by a piecewise linear homeomorphism in such a way that points of our cycle coincide with the sequence . In other words, without loss of generality we may assume that form a cycle for (topological entropy is maintained by topological conjugacy). Observe that the set is at most countable, since is piecewise linear. We can also embed intervals of sufficiently small diameters around points in (so that the total sum of these diameters is finite), similarly as it is done in the case of the standard Donjoy extension for circle rotation [3]. Each of these intervals is transformed from one onto another with the order defined by on . Entropy remains unchanged (homeomorphism on the interval has topological entropy zero) and, hence, there is no DC3 pair for our modified map. But now we have a periodic sequence of intervals for which were embedded in place of periodic orbit , and we may also assume that points are in the interiors of these intervals (if not, we use piecewise linear homeomorphism once again). Without loss of generality we may assume that a small neighborhood of has an invariant neighborhood on which is a homeomorphism.
Now it is enough to repeat the trick used in Theorem 3 in each of the intervals defined by points and the neighborhood of to produce discontinuities of the functions and , where is a point attracted by the cycle and by the fixed point . Obviously, we must prevent fluctuations of distance on intervals embedded around points to have exactly points of discontinuity of and .

It seems that the ideas of Theorem 4 can be extended even further. If instead of cycle we take an adding machine acting on the Cantor set properly embedded in and next arrange intervals along a dense orbit (exactly the same way as in Donjoy example [3]), then there is a hope that a pair with a countable set of discontinuities is constructed. In other words, it seems possible that the following question has a positive answer.

Question 1. Is there a map with zero topological entropy which has a pair such that is countable?

While no answer to the question raised earlier is provided, the following theorem shows that can be countable for a pair which is not DC3.

Theorem 5. There is a map and a pair such that is not DC3 but is infinite.

Proof. Put and endow it with the natural metric . It is well known that the shift map given by , , is continuous.
We are going to construct two special sequences . For and denote by the constant sequence consisting of exactly elements and by the infinite constant sequence . If are sequences, then denotes the concatenation of these two sequences, denotes the -times concatenation of with itself, and denotes the length of .
Put , , , for , and
Denote that . Then . For , denote that ; then
We put for all , so in particular .
Let and . Let be the union of closure of orbits of and . For by we denote the finite subsequence of formed by entries from th to th position, that is, if , where for , then .
Note that and ; hence, and so
Fix any and any . If is an index such that is a subblock of of the form , then by the structure of and , we have that is a subblock of . Assume that as abovementioned has been fixed. We have two possibilities.
Case A. is an even number. Then where Thus Since , we have .
Case B. is an odd number. Then where Thus Since , we have .

From Case A and Case B we can see that if is fixed and increases, then tends to .

Thus, provided that lies in of , the related and tend to as , and, hence,

Now we are ready for the main proof. For any positive number , we can write where ,  , and are uniquely determined. Thus where .

Case I. Fix any , .

By (17), when is large enough, we have that(a) if and block falls within or , then ,(b)similarly, if and block does not intersect blocks and in , then .

Let us denote that . Then, by (a) and (b) aforementioned, for large enough, we have that Additionally, if increases, then increases as well, and, hence,

Observe that Hence, by the previous calculations and (9), we see that But and so, finally, by (10) we obtain that In other words, we have just proved that for any and any .

It is not hard to verify that for every .

Case II. It remains to analyze the situation when for some . To estimate values of functions and , let us consider the particular case of ; that is, in (18).

Case C. Let be an even number. By (17) and the previous discussion of Case A with , if is large enough, then if and lies within some , then ;if and does not intersect block , then .

Thus performing calculations similar to these done in Case I leads to the following:

Case D. Let be an odd number. By (17) and the discussion in Case B, we have that when is large enough thenif and is a subblock of some , then ;if and does not intersect block , then .

Again, repeating calculations similar to these in Case I, we see that

Combining Cases C and D, we obtain that , provided that for some . Summing up Cases I and II together, we see that , which completes the proof.

Corollary 6. Let be the dynamical system defined in Theorem 5. Then it has no DC3 pair.

Proof. We can easily obtain from Theorem 5 that when and for some , , since
Observe that
Hence for any , we have the following: Each case is either trivial or has very similar proof which follows directly from Theorem 5 and (27).

2.2. Higher Iterates

It is well known that DC1 or DC2 pairs are preserved by higher iterates; that is, DC1 (or DC2) pair for is also DC1 (resp., DC2) pair for for every and vice versa. In this section we will show that there is no such correspondence in the case of DC3 pair.

Theorem 7 (see Theorem  1 of [10]). If is a DC3 pair of , then for every there is such that is DC3 for .

Proof. Let be a DC3 pair for , and let be an open interval such that for every . There is such that each function , , , is continuous at .
Observe first that if we denote that , for some and (here depend on ; i.e., , ), then This immediately implies that Similar calculations lead to the following: Now, if none of pairs is DC3, then for , and, hence, , which is a contradiction.

Next we show that not all are allowed choices in Theorem 7.

Theorem 8. There is a map and a pair such that is DC3 for but not DC3 for .

Proof. Put for . Then we have
Define a sequence by putting , for , and for . Next put for . Finally .
We define by putting , , , for and for . Since , it is easy to verify that is continuous.
Put and . By (33) and comparing with , we have that Thus for all . That is to say, is not a DC3 pair for .
On the other hand, for all ; hence, by (33) we have that Therefore and for every . In other words, we have just proved that is a DC3 pair for , and so the proof is completed.

Remark 9. In [10] there is another example of with such property as in Theorem 8. But the example here is more simple and has zero topological entropy.

Theorem 10. There is a map and a pair such that is DC3 for but not DC3 for .

Proof. Let be the sequences defined in Theorem 8 with the only difference that now if and , where , denote first coordinates of respective points. We keep as Euclidean distance for all other points. Now, if we consider as in Theorem 8, then distance between their orbits under is equal to for any iterate of , while , for every .


The research of Lidong Wang is supported by National Natural Science Foundation of China (no. 11271061). The main part of this paper was written during Piotr Oprocha's visit to the Department of Mathematical Sciences of the Dalian Nationalities University. His research leading to results contained in this paper was supported by the Marie Curie European Reintegration Grant of the European Commission under Grant Agreement no. PERG08-GA-2010-272297. He was also supported by the Polish Ministry of Science and Higher Education. Financial support of these institutions is widely acknowledged.


  1. 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
  2. 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
  3. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, USA, 2nd edition, 1989. View at MathSciNet
  4. H. Wang, G. Liao, and Q. Fan, “Substitution systems and the three versions of distributional chaos,” Topology and Its Applications, vol. 156, no. 2, pp. 262–267, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. F. Balibrea, J. Smítal, and M. Štefánková, “The three versions of distributional chaos,” Chaos, Solitons and Fractals, vol. 23, no. 5, pp. 1581–1583, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. T. Downarowicz, “Positive topological entropy implies chaos DC2,” In press.
  7. P. Oprocha, “Distributional chaos revisited,” Transactions of the American Mathematical Society, vol. 361, no. 9, pp. 4901–4925, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. P. Oprocha, “Weak mixing and product recurrence,” Annales de l'Institut Fourier, vol. 60, no. 4, pp. 1233–1257, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. Liao and Q. Fan, “Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy,” Science in China A, vol. 41, no. 1, pp. 33–38, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Dvořáková, “On a problem of iteration invariants for distributional chaos,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 785–787, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet