
1. Introduction
In this research paper, new types of minimal systems, transitivity and exactness are introduced and studied. This is intended as a survey article on transitivity and chaoticity of a discrete system given by θ-irresolute self-map of a topological space. On one hand, it introduces postgraduate students to the study of new types of exactness, minimal systems and chaotic maps and gives an overview of results on the topic, but, on the other hand, it covers some of the recent developments of dynamics, technology, electronic and computer science. We denote the interior and the closure of a subset A of X by
and
respectively. By a space X, we mean a topological space
. A point
is called a θ-adherent point of A [1] , if
for every open set U containing x. The set of all θ-adherent points of a subset A of X is called the θ-closure of A and is denoted by
. A subset A of X is called θ-closed if
. Dontchev and Maki [2] have shown that if A and B are subsets of a space X, then
and that
. Recall that a space
is Hausdorff if and only if every compact set is θ-closed. The complement of a θ-closed set is called a θ-open set. The family of all θ-open sets forms a topology on X and is denoted by
. This topology is coarser than τ and that a space
is regular if and only if
[3] . Note also that the θ-closure of a given set needs not be a θ-closed set. Our purpose is to investigate some new types of transitivity, because when we confirm not existence of θ-transitivity we can’t confirm the existence of other types of transitive functions and then we can’t study chaos theory, i.e. theta-transitivity absence cannot find other types of transitive maps therefore cannot find chaotic maps. For more knowledge about transitivity and chaotic maps see references [4] and [5] .
2. Preliminaries and Definitions
Definition 2.1
A point
is said to be A θ-interior point of A, if there exists an open set U containing x such that
.
The set of all θ-interior points of A is said to be the θ-interior of A, and is denoted by
. It is obvious that an open set U in X is θ-open if
.
Definition 2.2
1) A map
is a homeomorphism if it is continuous, bijective and has a continuous inverse.
2) A map
is θr-homeomorphism if it is bijective and thus invertible and both h and h−1 are θ-irresolute.
3) The systems
and
are topologically conjugate or conjugate if there is a homeomorphism
such that
4) The systems
and
are topologically θr conjugate or θr conjugate if there is θr-homeomorphism
such that
Definition 2.3 A map f is said to be transitive (resp., θ-transitive [6] ) if for any non-empty open (resp., θ-open) sets U and V in X, there exists
such that
.
Kaki definition 2.4 A map f is said to be 1-transitive, if for every
, there exists
such that
.
Theorem 2.5 every 1-transitive implies transitive.
Proof:
We have to prove that for any non-empty open sets U and V in X, there exists
such that
. Now, since U and V are non-empty so there is
and
so there is
such that
since f is 1-transitive and
but
, we have
i.e.
.
Definition 2.6
The points x is called a non-wandering point if for every open set U containing x there is an integer
such that
.
Definition 2.7
The non-wandering set of a map f,
, includes the points x such that for every open set U containing x there is an integer
such that
.
Definition 2.8
The points x is called theta-non-wandering point if for every θ-open set U containing x there is an integer
such that
.
Definition 2.9
The theta-non-wandering set of a map f,
, includes the points x such that for every θ-open set U containing x there is an integer
such that
.
Proposition 2.10 Every non-wandering point is a theta-nonwandering point but not conversely.
3. Action of a Group on a Topological Space
If G is a group and X is a topological space, then a group action φ of G on X is a function
such that
that satisfies the following three axioms [7] and [8]:
1)
is continuous, for all g in G
2) Identity:
for all x in X. (Here, e denotes the neutral element of the group G.)
3) Compatibility
for all g, h in G and all x in X.
The group G is said to act on X (on the left). The set X is called a (left) G-set.
Definition 3.1
1) The action of G on X is called 1-transitive if X is non-empty and if for each pair x, y in X there exists a g in G such that
.
2) The action of G on X is called topologically-transitive if X is non-empty and if for each non-empty pair
, there exists a g in G such that
.
Theorem 3.2 Every 1-transitive implies topologically-transitive.
Proof: The same technique of theorem 2.5.
Definition 3.3 The action of G on X is called n-transitive if X is non-empty and if for any two ordered sets of n different points
and
in X there exists a g in G such that
for
.
Definition 3.4 The action of G on X is called topologically n-transitive if X is non-empty and if for each non-empty pair
, such that
and
for
there exists a g in G such that
for
.
Definition 3.5 Relative to
a point
is called wandering if there is an open U containing x such that
for all
.
Definition 3.6
1) A point
is θ-recurrent if, for every θ-open set U containing x, infinitely many
satisfy
.
2) Let X be a topological space,
be θ-irresolute map, then the function g is called topologically θ-strongly mixing if, given any nonempty θ-open subsets
such that
for all
.
3) A subset B of X is g-invariant if
. A non-empty θ-closed invariant subset B of X is θ-minimal, if
for every
. A point
is θ-minimal if it is contained in some θ-minimal subset of X
Theorem 3.7 if g is topologically θ-strongly mixing then it is also θ-transitive but not conversely.
Definition 3.8
1) The function f is θ-exact if, for every nonempty θ-open set U, there exists some
such that
.
2) The function g is (topological) θ-transitive (resp., θ-mixing) if for any two nonempty θ-open sets
, there exists some
such that
. (resp.
for all
).
3) The function g is weak θ-mixing if
is θ-transitive on
4) The θ-mixing function
is pure θ-mixing if and only if there exists θ-open set
such that
.
5) The function
is θ-chaotic if g is θ-transitive on X and the set of periodic points of g is θ-dense in X.
6) The function g is called θ-exact chaos (resp., θ-mixing chaos and weakly θ-mixing chaos) if g is θ-exact (resp., θ-mixing and weakly θ-mixing) and θ-chaotic function on the space X.
Theorem 3.9 Theorem 3.9 topological θ-exactness implies θ-mixing implies weakly θ-mixing implies θ-transitivity
Theorem 3.10 If
and
are topologically θr-conjugate by a θr-homeomorphism
. Then
1) The map f is θ-exact if and only if g is θ-exact
2) The map f is θ-mixing if and only if g is θ-mixing
3) The map f is θ-chaotic if and only if g is θ-chaotic
4) The map f is weakly θ-mixing if and only if g is weakly θ-mixing.
Theorem 3.11 if
and
are θr-conjugate via
. Then
1) T is θ transitive subset of X
is θ-type transitive subset of Y;
2) T is θ-mixing subset of X
is θ-mixing subset of Y.
Corollary A subset B in the space X is a chaotic set
is a chaotic set in the space Y.
Proof (1)
Assume that
and
are topological systems which are topologically θr-conjugated by
. Thus, h is θr-homeomorphism (that is, h is bijective and thus invertible and both h and h−1 are θ-irresolute) and
Suppose T is θ-type transitive subset of X. Let A, B be θ-open subsets of Y with
and
(to show
for some
).
and
are θ-open subsets of X since h is an θ-irresolute.
Then there exists some
such that
since T is θ-type transitive subset of X, with
and
. Thus (as
implies
).
Therefore,
implies
since h−1 invertible. So h(T) is θ-type transitive subset of Y.
Proof (2)
We only prove that if T is topologically θ-mixing subset of Y then h−1(T) is also topologically θ-mixing subset of X. Let U, V be two θ-open subsets of X with
and
. We have to show that there is N > 0 such that for any
,
.
and
are two θ-open sets since h is θ-irresolute with
and
. If the set T is topologically λ-mixing then there is N > 0 such that for any n > M,
. So
. That is
and
for
.
. Thus, since
, so that
and we have
that is
. So, h−1(T) is θ-mixing set.
Theorem 3.12 Let
be a topological system and A be a nonempty θ-closed set of X. Then the following conditions are equivalent.
1) A is a θ-transitive set of
.
2) Let V be a nonempty θ-open subset of A and U be a nonempty θ-open subset of X with
. Then ther exists
such that
.
3) Let U be a nonempty θ-open set of X with
. Then
is θ-dense in A.
Theorem 3.13 Let
be topological system and A be a nonempty θ-closed invariant set of X. Then A is a θ-type transitive set of
if and only if
is θ-type transitive set.
Proof:
) Let
and
be two nonempty θ-open subsets of A. For a nonempty θ-open subset
of A, there exists a θ-open set U of X such that
Since A is a θ-type transitive set of
, there exists
such that
. Moreover, A is invariant, i.e.,
, which implies that
. Therefore,
, i.e.
. This shows that
is θ-type transitive.
) Let
be a nonempty θ-open set of A and U be a nonempty θ-open set of X with
, Since U is an θ-open set of X and
, it follows that
is a nonempty θ-open set of A. Since
is topologically θ-type transitive, there exists
such that
, which implies that
. This shows that A is a θ-type transitive set of
.
We now introduce four definitions of dense orbit and transitivity as follows:
Definition 3.14 Let X be a topological space.
1) A map f is said to have θ-dense orbit if there exists
such that
;
2) A map f is said to have strictly dense orbit (resp., strictly θ-dense orbit), if there exists
such that
(resp.
);
3) A map f is said to be ω-transitive if there exists
such that
;
4) A map f is said to be transitive (resp., θ-transitive) if for any non-empty open (resp., θ-open) sets U and V in X, there exists
such that
.
Now, we will discuss the relations between these orbits and transitivity maps in the theorem below:
Theorem 3.15
1) For any topological space X, each strictly θ-dense orbit is α θ-dense orbit, and each ω-transitive map has strictly dense orbit and it is transitive.
2) For any topological space X, each continuous map which has strictly dense orbit is ω-transitive and transitive.
Acknowledgements
First, thanks to my family for having the patience with me for having taking yet another challenge which decreases the amount of time I can spend with them. Specially, my wife who has taken a big part of that sacrifice, and also Sarmad, my son who helps me for typing my research. Thanks to all my colleagues for helping me for completing my research.