Sensitivity of the Product System of Set-Valued Non-Autonomous Discrete Dynamical Systems ()
1. Introduction
Since the beginning of the 21st century, the problem of chaos in set-valued discrete systems has been discussed warmly. In 2003, Román-Flores [1] studied the interaction of transitivity between
and its induced system
. Following his work, many scholars studied the chaotic properties of set-valued discrete systems. For example, transitivity, mixing, Kato’s chaos (see [2] - [9] and others). About sensitivity, Liu, Shi, and Liao [10] proved that if
is a surjective, continuous interval map, then
is sensitive if and only if
is sensitive. And in 2013, they gave an example to show that Li-Yorke sensitivity of
does not necessarily imply Li-Yorke sensitivity of
( [11] ). In 2010, Sharma and Nagar [12] showed the relations between the various forms of sensitivity of the systems
and
and proved that all forms of sensitivity of
partly imply the same for
. In 2014, Yang, Wang, and Li [13] studied the relationship between non-autonomous dynamical system and its hyperspace system in the aspect of sensitivity. In 2015, Wu, Wang, and Chen [14] had obtained a few sufficient and necessary conditions to ensure a dynamical system be
-sensitive or multi-sensitive.
Inspired by the literature [13] - [17], this paper further studies some stronger forms of sensitivity in set-valued discrete systems. The structure of this paper is as follows. In Section 2, some basic definitions are given. In Section 3, the main results are established and proved.
2. Preliminaries
2.1. Non-Autonomous Discrete System
In this paper, let
, and the metric on X is denoted as
.
is a mapping sequence, and denoted by
. This sequence defines a non-autonomous discrete system
. Under this mapping sequence, the orbit of the point
is
, where
,
denotes the identity mapping. Similarly,
.
2.2. Set-Valued Product Systems
Let
be the hyperspace on X. That is, the space of nonempty compact subsets of X with the Hausdorff metric
defined by
for any
. Clearly,
is a compact metric space. The system
induces a set-valued dynamical system
, where
is defined as
for any
. For any finite collection
of nonempty subsets of X, let
where the topology on
given by the metric
is the same as the Vietoris or finite topology, which is generated by a basis consisting of all sets of the following form:
, where
is an arbitrary finite collection of nonempty open subsets of X.
Two set-valued non-autonomous discrete systems
and
are defined in compact metric spaces
and
, whose metrics are
and
. Let
, define
, where
. The metric of
is
, and define
, where
. Then the system
is called the product dynamical system of the two set-valued non-autonomous discrete systems.
2.3. Definitions of Sensitivity
In this section, some definitions of the sensitivity in set-valued discrete systems will be given.
Definition 1. ( [14] ) Let
be the collection of all subsets of
. A collection
is called a Furstenberg family if it is hereditary upwards, i.e.,
and
imply
. A family
is proper if it is a proper subset of
, i.e. neither empty nor the whole
.
Definition 2. ( [15] ) Let X be a nonempty set and F be a family of nonempty sets composed of subsets of X, if
1)
then
; 2)
and
then
, then F is said to be a filterdual on the set X.
Definition 3. ( [9] ) Given
, its upper and lower densities are defined by
If
, then the set A has density
. Let
, where
is the collection of all infinite subsets of
, then
is a Furstenberg family. It can be verified that
This implies that
is a filterdual.
Definition 4. ( [18] [19] [20] [21] [22] ) Let
be a system and let
be a Furstenberg family.
1)
is sensitive, if there exists an
such that for any
and any
, there exist
with
such that
;
2)
is infinitely sensitive, if there exists an
such that for any
and any
, there exist
with
such that
;
3)
is
-sensitive, if there exists an
such that for any
and any
, there exist
with
such that
;
4)
is
-sensitive with the sensitive constant
if for any
and any
, there exist
with
such that
and
for any
;
5)
is ergodically sensitive if there exists
(ergodically sensitive constant) such that for any nonempty open subset
,
. Where
is the diagonal of the set.
3. Main Results
Based on the definitions in Section 2, we now further investigate the dynamical properties of the product systems.
Lemma 1. ( [14] ) Let
be two compact metrics. Then, for any
and any
, there exist nonempty open subsets
and
such that
.
Theorem 1. Let
and
be two set-valued non-autonomous discrete systems, then
is sensitive if and only if
or
is sensitive.
Proof. (Necessity) Assuming that
and
are not sensitive, it is proved that
is not sensitive.
1) If
is not sensitive, then for any
, there exist
and
, there is
for any
satisfying that
2) If
is not sensitive, then for any
, there exist
and
, there is
for any
satisfying that
Since
, then there is
for any
, one has that
So
Let
, then
Therefore,
is not sensitive and contradicts the proposition. So
or
is sensitive.
(Sufficiency) For any nonempty open set
, by Lemma 1, there exist nonempty open subsets
and
such that
If
is sensitive, then there exists
, for any
and
, there is
such that
, one has that
Take any point
in
, where
. Define
,
,
,
. Therefore,
Then for any
, there exists
such that
So
Hence,
is sensitive.
Theorem 2. Let
and
be two set-valued non-autonomous discrete systems, then
is infinitely sensitive if and only if
or
is infinitely sensitive.
Proof. (Necessity) Assuming that
and
are not infinitely sensitive, it is proved that
is not infinitely sensitive.
1) If
is not infinitely sensitive, then for any
, there exist
and
, there is
for any
satisfying that
2) If
is not infinitely sensitive, then for any
, there exist
and
, there is
for any
satisfying that
Since
, then there is
for any
, one has that
So
Let
, then
Therefore,
is not infinitely sensitive and contradicts the proposition. So
or
is infinitely sensitive.
(Sufficiency) For any nonempty open set
, we know from Lemma 1, there exist nonempty open subsets
and
such that
If
is infinitely sensitive, then there exists
, for any
and
, there is
such that
, one has that
Define
are the same as Theorem 3.1, then
So
Therefore,
is infinitely sensitive.
Theorem 3. Let
and
be two set-valued non-autonomous discrete systems, then
is
-sensitive if and only if
or
is
-sensitive.
Proof. (Necessity) Assuming that
and
are not
-sensitive, it is proved that
is not
-sensitive.
1) If
is not
-sensitive, then for any
, there exist
and
, there is
for any
satisfying that
2) If
is not
-sensitive, then for any
, there exist
and
, there is
for any
satisfying that
Since
, then there is
for any
, one has that
So
Let
, then
Therefore,
is not
-sensitive and contradicts the proposition. So
or
is
-sensitive.
(Sufficiency) For any nonempty open set
, we know from Lemma 1, there exist nonempty open subsets
and
such that
If
is
-sensitive, then there exists
, for any
and
, there is
such that
, one has that
By the same proof as Theorem 3.1, one can obtain that
Combining this with the hereditary upwards property of
, it follows that
Hence,
is
-sensitive.
Theorem 4. Let
and
be two set-valued non-autonomous discrete systems, if
is
-sensitive then
or
is
-sensitive.
Proof. Assuming that
and
are not
-sensitive, it is proved that
is not
-sensitive.
1) If
is not
-sensitive, then there is
, there exist
and
, there is
for any
satisfying that
2) If
is not
-sensitive, then there is
, there exist
and
, there is
for any
satisfying that
Since
, then there is
for any
, one has that
So
Let
, then
Therefore,
is not
-sensitive and contradicts the proposition. So
or
is
-sensitive.
Lemma 2. ( [14] ) Let
be a non-autonomous dynamical system, for any nonempty open set
and any
, one has
.
Theorem 5. Let
and
be two non-autonomous dynamical systems and let
be a Furstenberg family such that
is a filterdual. If
is ergodically sensitive, then
or
is ergodically sensitive.
Proof. Suppose that both
and
are not ergodically sensitive. Then, for any
, there exist nonempty open subsets
and
such that
and
So
and
Since
is a filterdual, then
. For any
, noting that
and
, applying Lemma 3.2, one has that
So
This implies that
Since
is arbitrary, it shows that
is not ergodically sensitive. Therefore,
or
is ergodically sensitive.
4. Conclusion
For set-valued non-autonomous discrete dynamical systems, the sensitivity of the product systems and the factor systems are consistent most of the time, for example, sensitive, infinitely sensitive,
-sensitive, and
-sensitive, while it is not a necessary and sufficient condition for ergodically sensitive. Based on this paper and others, one can further consider some other chaotic properties of set-valued discrete systems, retention conditions for compound operation, and so on, which are worthy of study.
Acknowledgements
This work was funded by the Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (No. 2020WZJ01), the Scientific Research Project of Sichuan University of Science and Engineering (No. 2020RC24), and the Graduate student Innovation Fund (Nos. y2020077, y2021100)