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We introduce and study some concepts of sensitivity via Furstenberg
families. A dynamical system

Throughout this paper a topological dynamical system (TDS) is a pair

The phrase—sensitive dependence on initial condition—was first used by Ruelle [

Recently, several authors studied the sensitive property (cf. Abraham et al. [

Let

There exists a positive

There exists a positive

There exists a positive

From Proposition

Some authors introduced concepts which link the Li-Yorke versions of chaos with the sensitivity in the recent years. Blanchard et al. [

Akin and Kolyada [

We see that Li-Yorke sensitivity clearly implies spatiotemporal chaos, but the latter property is strictly weaker (see [

Let

where

A pair

Let

In the past few years, some authors [

In this paper we investigate the sensitivity from the viewpoint of Furstenberg families.

A dynamical system

A dynamical system

In Section

In this section, we introduce some basic notions related Furstenberg families (for details see [

Clearly, if

A Furstenberg family

A Furstenberg family

For Furstenberg families

For every

Clearly,

Let

In particular we have

Let

Let

Let

In this section, we introduce and study the concept of

We will use the following relations on

For any subset

We define the sets of

We say that

We say that

Let

If

Let

Let

Suppose that

Hence,

Let

Since

The following lemma is proved in [

Let

(1) Let

Hence,

(2) Suppose that

Suppose that

By Lemma

Let

there exists a positive

Since

Hence

The following theorem is based on arguments in Huang and Ye [

Let

There exists a positive

There exists a positive

There exists a positive

There exists a positive

(1)

(2)

(3)

(1)

Thus, we have proved that (1)–(3) are equivalent.

(4)

(3)

(2)

Let

Since

A map is semiopen if the image of an opene subset contains an opene subset. A factor map

Let

Let

Let

In this section, we introduce and study the notion of (

Let

We denote the set of all

The following lemma comes from [

Let

Let

Note that [

Suppose that

A TDS

When

A TDS

That is, for all

If the pair

We will use the following lemmas which comes from [

Let

Let

Let

Since

Let

Suppose that

Let

As

Finally, as examples we will discuss the

Let

We define a metric

Let

Let

Suppose that

By Lemma

Now we show that

Suppose that

Let

By Lemmas

Suppose that

The authors greatly thank the referees for the careful reading and many helpful remarks. This work was supported by the National Nature Science Funds of China (10771079, 10471049), and Guangzhou Education Bureau (08C016).