DC 3 Pairs and the Set of Discontinuities in Distribution Functions

In 1994, Schweizer and Smı́tal extended the definition of LiYorke pair for intervalmaps [1].Themainmotivationwas 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 (X, f) is a dynamical system, that is, a continuous map f : X → X acting on a compact metric space (X, d) (basic definitions related to dynamical systems, such as orbit and ω-limit set, can be found in any standard book on dynamical systems, e.g., [3]). For any positive integer n, points x, y ∈ X, and real number t > 0, let


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.
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 2) 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 1.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.

Distributional Chaos of Type 3
In this section we will focus on properties of distribution functions Φ  and Φ *  , which may cause many problems during numerical investigation of the dynamics.

Discontinuity Points.
The essential ingredient of all the three definitions of DC pairs is "sufficiently large" difference in values of functions Φ  and Φ *  .Accordingly, an important question is how much values of Φ  and Φ *  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 As we see, that Φ  () ̸ = Φ *  () 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 Φ  () or Φ *  (), 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, (, ) ∩ Q 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   .
Proof.Fix any increasing sequence   > 0 such that lim  → ∞ (  / +1 ) = 0. Let {  } be a decreasing sequence such that 0 <   < min{1 − , } for all  ≥ 0. 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  ∈ [0,1] its -limit set is the singleton consisting of one of the points 0, , 1. Similarly we can verify that Ω() = {0, , 1} and so  has zero topological entropy.Then it has no DC3 pair by [1].
We can see that Therefore we can easily verify that Thus Φ *  () = 1 and Φ  () = 0. On the other hand, for any We can extend the construction in Theorem 3 to the following.

Theorem 4. For any finite sequence
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 2  elements for some  > 0. Let  be a piecewise linear map of type 2  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 2  elements contained in the interval (0, 1).We may assume that 0, 1 are fixed points of .We can also transform the interval [0, 1] by a piecewise linear homeomorphism in such a way that points of our cycle coincide with the sequence  1 < ⋅ ⋅ ⋅ <  2  .In other words, without loss of generality we may assume that  1 , . . .,  2  form a cycle for  (topological entropy is maintained by topological conjugacy).Observe that the set  = { ∈ [0, 1] ;   () =   for some  ≥ 0 and 1 ≤  ≤ } (7) 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  1 , . . .,  2  , 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 0 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 0 to produce discontinuities of the functions Φ *  and Φ  , where  is a point attracted by the cycle  1 , . . .,  2  and  by the fixed point 0. Obviously, we must prevent fluctuations of distance on intervals embedded around points  +1 , . . .,  2  to have exactly  points of discontinuity of Φ *  and Φ  .
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 (0, 1) 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  : [0, 1] → [0, 1] 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 (, ) It is well known that the shift map  : Y → Y given by ()  =  +1 ,  = 1, 2, . .., is continuous.
Case II.It remains to analyze the situation when  = 1/2  for some  ≥ 0. To estimate values of functions Φ  () and Φ *  (), let us consider the particular case of  =   ; that is,  =  = 0 in (18).

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  > 1 and vice versa.In this section we will show that there is no such correspondence in the case of DC3 pair.
Theorem 8.There is a map  and a pair (, ) such that (, ) is DC3 for  2 but not DC3 for .