Abstract

In this paper, we have study some concepts of minimal open, closed sets and minimal functions. Further, we have shown that these properties preserved under conjugate maps.

1. Introduction

Let be a compact topological space. All maps under consideration are supposed to be continuous. The set of all continuous maps f: X → X is denoted by C(X). By a system we mean a compact topological space (phase space) X and In a topological space a trajectory consists of a sequence of points and can possibly contain additional attributes a measured at each point. Trajectories can be generated by moving objects but also by moving phenomena, e.g. measurement points on a hill slide. The pointscan be captured at regular intervals or irregularly A point “moves,” its trajectory ¹ being the sequence where is the nth iteration of The point is the position of the point x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by. A map f ∈ C(X) is (topologically) transitive if for any two nonempty open sets U and V in X, there is a nonnegative integer n such that . If X has no isolated points then this definition is equivalent to the existence of a dense orbit, i.e. . If every orbit of f is dense, the map f is called minimal. Denote by the set of transitive self-maps of the space X. A minimal map is necessarily surjective if is assumed to be Hausdorff and compact.

Now, to study the existence of minimal sets, given a system a set is called a minimal set if it is non-empty, closed and invariant and if no proper subset of has these three properties. So, is a minimal set if and only if is a minimal system. A system is minimal if and only if X is a minimal set in The basic fact discovered by G. D. Birkhoff is that in any compact system there are minimal sets. This follows immediately from the Zorn's lemma. Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. This is how compact minimal sets may appear in non-compact spaces. Two minimal sets in either are disjoint or coincide. A minimal set A is strongly i.e. Provided it is compact Hausdorff

2. Preliminaries and Definitions

Definition 2.1

1. (Minimal Hausdorff spaces) ³

A topological space is said to be minimal Hausdorff if is Hausdorff and there exists no Hausdorff topology on X strictly weaker than Thus this minimality property is topological.

2. Two topological spaces and are called homeomorphic ⁵ if there exists a one-to-one onto function such that and are both continuous.

3. Two topological systems and are said to be topologically conjugate if there is a homeomorphism such that We will call h topological Conjugacy. Thus, the two topological systems with their respective function acting on them share the same dynamics(see the following diagram )

Definition 2.2 (minimal open set). Recall that a proper non empty open subset U of a topological space X is said to be a minimal open set, if any open set which is contained in U is empty set or U.

Definition 2.3 Recall that a proper non empty closed subset F of a topological space X is said to be a minimal closed set if any closed set which is contained in F is empty set or F.

Definition 2.4

A system is called minimal if does not contain any non-empty, proper, closed subset. In such a case we also say that the map itself is minimal. Thus, one cannot simplify the study of the dynamics of a minimal system by finding its non- trivial closed subsystems and studying first the dynamics restricted to them.

Proposition 2.5

Let be continuous function. The following are equivalent:

1. is minimal.

2. The only closed invariant sets of X are X itself and the empty set.

3. For any non-empty open subset then.

Proof:

(1) (2)

Suppose that is a non-empty closed invariant set. Let Then since C is invariant, Since C is closed, so, but since the orbit is dense, this means . Thus X=C.

(2) (3), let be a non-empty open set. Put then C is closed and invariant. Since by (2) we must have

(3) (1), let and let U be an arbitrary non-empty open subset. Then by (3), for some . Thus , and hence Since U was arbitrary, is dense, i.e. f is minimal.

Definition 2.6 (minimal) Let X be a topological space

And be continuous map on X. Thenis called minimal system (or f is called minimal map on X) if one of the three equivalent conditions hold:

(1) The orbit of each point in X is dense in X

(2) for each x X.

(3) Given x X and a nonempty open U in X, there exists such that

Definition 2.7 A subset M of X is said to be minimal under provided that M is non-empty, closed and invariant, that is and no proper subset of M has all these properties.

Theorem 2.8 ⁴ Any two minimal sets must have empty intersection.

Proof: Let be two distinct minimal sets, and suppose that. Then A is closed, and fore very a ∈ A and every n ∈ N, , so A is invariant. But then A is a proper subset of both and which is closed, invariant and non-empty, contradicting the fact that are minimal.

Definition 2.9 Two topological systems and are said to be topologically conjugate if there is a homeomorphism such that. We will call h a topological Conjugacy.

We have stated a new proposition as follows:

Proposition 2.10 if are topologically conjugated by . Then is a minimal open set in X if and only ifis a minimal open set in Y.

Definition 2.11 Let X be a topological space, and a continuous map. We say f is (topologically) transitive if for any nonempty open sets there exists n > 0 such that . We say f is strongly transitive ³ if for any nonempty open set for some s > 0. For more knowledge see ⁴.

Definition 2.12 A subset S of X is called -set if it is the intersection of open sets containing S.

Definition 2.13 Recall that a subset of a topological space is called λ-closed set if where S is -set and C is a closed set.

Proposition 2.14 Let be topological space and A be a nonempty λ-closed invariant set of X. Then A is a λ-type transitive set of if and only if is λ-type transitive.

Proof:

Let be two nonempty λ-open subsets of A. For a nonempty λ-open subset of A, there exists a λ- open set U of X such that Since A is a λ-type transitive set of , there exists n ∈ N such that Moreover, A is invariant, i.e., which implies that Therefore, , i.e. . This shows that is λ- type transitive.

Let be a nonempty λ-open set of A and U be a nonempty λ-open set of X with Since U is an λ-open set of X and , it follows that U ∩ A is a nonempty λ-open set of A. Since is topologically λ-type transitive, there exists n ∈ N such that which implies that. This shows that A is a λ-type transitive set of

Theorem 2.15 Let X be a non-empty λ-compact Hausdorff space. Then the intersection of a countable collection of λ-open λ-dense subsets of X is λ -dense in X.

Definition 2.16 Let be a topological space. Recall that subset A of X is called λ -dense in X if λCl(A)=X.

Corollary 2.17 A subset A of a space is λ -dense if and only if for all other than.

Proof: If A is λ -dense set in X, then by definition, , and let U be a non-empty λ-open set in X. Suppose that A∩U=ϕ. Therefore is λ-closed and So, , i.e. , but , so XB, this contradicts U ≠ φ.

Theorem 2.18 Let be a non-empty λ-compact Hausdorff space and is λ -irresolute map and that X is λ-type separable. Suppose that f is topologically λ -type transitive. Then there is an element such that the orbit is λ -dense in X.

Proof: Let be a countable basis for the λ -topology of X. For each i, let for some

Then, clearly is λ -open and λ -dense. It is λ -open since f is λ -irresolute, so, is λ - open and λ -dense since f is topological λ- transitive map. Further, for every λ -open set V, there is n>0, such that since f is λ- transitive.

Now, apply theorem 2.15 to the countable λ -dense set {} to say that is λ -dense and so non-empty. Let . This means that, for each i, there is a positive integer n such that for each i. By corollary 2.17, this implies that is λ -dense in X.

References