^{1}

^{2, 3}

^{4}

^{1}

^{2}

^{3}

^{4}

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.

In 1994, Schweizer and Smítal extended the definition of Li-Yorke pair for interval maps [

Let us first give the concepts that originated from [

Suppose that

If a pair of points

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 [

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.

In this section we will focus on properties of distribution functions

The essential ingredient of all the three definitions of DC pairs is “sufficiently large” difference in values of functions

The following conditions are equivalent:

Implication (2)

Conversely, assume that

As we see, that

For any set

For any

Fix any increasing sequence

Consider the pair

We can see that

We can extend the construction in Theorem

For any finite sequence

We only sketch the idea of this construction, leaving exact calculations to the reader.

First, extend the sequence

Now it is enough to repeat the trick used in Theorem

It seems that the ideas of Theorem

Is there a map

While no answer to the question raised earlier is provided, the following theorem shows that

There is a map and a pair

Put

We are going to construct two special sequences

Put

Denote that

We put

Let

Note that

Fix any

From Case A and Case B we can see that if

Thus, provided that

Now we are ready for the main proof. For any positive number

By (

if

similarly, if

Let us denote that

Observe that

It is not hard to verify that

if

if

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

if

if

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

Combining Cases C and D, we obtain that

Let

We can easily obtain from Theorem

Observe that

Hence for any

It is well known that DC1 or DC2 pairs are preserved by higher iterates; that is, DC1 (or DC2) pair for

If

Let

Observe first that if we denote that

Next we show that not all

There is a map

Put

Define a sequence

We define

Put

On the other hand,

In [

There is a map

Let

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.