Almost periodic points and minimal sets in topological spaces

This Paper we characterize in this paper, we need the following terminology and concepts. Let N={0,1,2,…,} be the set of natural numbers and let Z={0,±2…,} be the set of integers. For a set A, |A| denotes the cardinality of the set A. If f :X→X is map (=continuous function) of a topological space X, then f0=Id (n≥1) denotes the composition with itself n times. The orbit of a point x∈X under f, denoted by 0+(x, f), is the set {fn├ (x)┤|n∈N}. Also if f: X→X is a homeomorphism, then we put f-n=(f-1)n(n≥1), where f-1 is the inverse of f. The two-sided orbit of a point x∈X under f, denoted by O+(x,f), is the set {fn├ (x)┤|n∈Z}. A point x∈X is called a periodic point of f if there exists a positive number N∈N such that fN(x)=x. A point x∈X is called an almost periodic point of f provided that for any neighborhood U of x in X, there exists N∈N such that {fn+i├ (x)┤|i=0,1,2,.........N}∩U≠ for all N∈N. We denote the set of all almost periodic points of f by AP(f). A subject W of X is invariant of f if W ≠and f(W)⊆W.A subset W of X is a minimal set of f if W is a closed invariant set off and W does not contain any proper closed invariant set off. A map f: X→X is minimal if X is a minimal set off. It is well known that if f: S1→S1 is an irrational rotation of the unit circle SI, then f is minimal.

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Introduction

A topological space X is a T1-space if for any distinct point x and y in X, there exist open sets U and V of X such that x∈ X, y∈ V, y U and xV.A topological space X is a Housdorif space if for any distinct points x and y in X, there exist disjoint open sets U and V of X such that x∈ X U and y∈ V. A topological space X is regular space if for any closed subset W of X, any point x∈ X -W, there exist disjoint open sets U and V such that x∈ U and W ⊆ V. A topological space x is an !-regular space if for any closed subset W of X, any point x∈ X -W and any countable subset A of W, exist disjoint open sets U and V such that x∈ U and A ⊆ V [1,2,3] . Now we assert and prove the main theorem on compact minimal set in topological dynamics.

Theorem:-Let E be a compact minimal set in X. The set E is almost periodic minimal if and only if the flow 𝜋 is equicontinuous on E.

Proof:-

Let us assume E be almost periodic minimal set and let E=H(x), where x is and almost periodic point. We claim that the flow 𝜋 is equicontinuous on E. We first show that 𝜋 is equicontinuous on 𝛾(x). Let an index 𝑎 ∈ A be preassigned and select b ∈ A, so by the uniformity condition if follows that 𝑋 𝑖 ∈ 𝑉 𝑏 (𝑥 𝑖+1 ), 𝑖 = 1,2,3.

Let L.0be an inclusion interval for the relatively dense set E (𝑏, 𝜋𝑥). Since the continuity of 𝜋 restricted to the compact set Ex [0,L] is uniform, for every b ∈ A there exists and index c such that for 𝑦, 𝑧 ∈, 𝑦 ∈ 𝑉 𝑐 (𝑧) implies that,

In order to show that π is equicontinuus on 𝛾(𝑥) for any tow points 𝜋(𝑥, 𝑡 1 ) and 𝜋𝜋(𝑥, 𝑡 2 ) of 𝑦(𝑥), 𝜋(𝑥, 𝑡 1 + 𝑡) ∈ V a (𝜋(𝑥, 𝑡 2+ )) whenever 𝜋(𝑥, 𝑡 1 ) ∈ V c (𝜋(𝑥, 𝑡 2 )) for all 𝑡 ∈ 𝑅. Let t ∈ 𝑅 be fixed. Then there exists a T in E(b,𝜋𝑥) such that 0 ≤ 𝜏 + 𝑡 ≤ 𝐿. From the definition of b-displacement of 𝜋𝑥, it follows that, 𝜋(𝑥, 𝜏 + 𝑡 + 𝑡 1 ) ∈ V b (𝜋(𝑥, 𝑡 1 + 𝑡))and 𝜋(𝑥, 𝜏 + 𝑡 1 + 𝑡) ∈ 𝑉 𝑏 (𝜋(𝑥, 𝑡 2 + 𝑡)) Applying the condition (2) of uniform continuity one has 𝜋(𝑥, 𝑡 1 + 𝑡) ∈ 𝑉 𝑏 (𝜋(𝑥, 𝑡 2 + 𝑡) Hence finally, by uniformity condition (1) we have 𝜋(𝑥, 𝑡 1 + 𝑡) ∈ 𝑉 𝑎 (𝜋(𝑥, 𝑡 2 + 𝑡)) i.e., the equicontinuity of 𝜋 on γ(𝑥) follows.

In order to show that 𝜋 is equicontinuos on E, let a ∈ A be any index. Then there b ∈ A in consistent with the uniformity condition (i). By using the equicontinuity of 𝜋 on 𝑦(𝑥) we can choose C∈A such that 𝜋(𝑥, 𝑡 1 + 𝑡)) ∈ 𝑉 𝑏 (𝜋(𝑥, 𝑡 2 𝑡)) whenever 𝜋(𝑥, 𝑡 1 ) ∈ V c (𝜋(𝑥, 𝑡 2 )) Now let us choose d so that 𝑥 𝑖 ∈ V 𝑑 (𝑥 𝑖 + 1), 𝑖 = 1,2,3

We will now show that if y, z are points in E=H (x) with 𝑦 ∈ 𝑉 𝑑 (𝑧), then 𝜋 𝑡 (𝑦) ∈ 𝑉 𝑎 (𝜋 𝑡 (𝑧)), 𝑖. 𝑒. , 𝜋(𝑦, 𝑡) ∈ 𝑉 𝑎 (𝜋(𝑧, 𝑡)). Since 𝑦, 𝑧 ∈ 𝐸 = 𝐻(𝑥), there are nets {𝜋(𝑥, 𝑡 𝑛 )} 𝑎𝑛𝑑 𝜋(𝑥, 𝑆 𝑛 )} in 𝑦(𝑥) with 𝜋(𝑥, 𝑡 𝑛 ) → 𝑦 𝑎𝑛𝑑 𝜋(𝑥, 𝑠 𝑛 ) → 𝑧 without the loss of generality, 𝜋(𝑥, 𝑡 𝑛 ) ∈ 𝑉 𝑑 (𝑦)𝑎𝑛𝑑 𝜋(𝑥, 𝑠 𝑛 ) ∈ 𝑉 𝑑 (𝑧). Since 𝑦 ∈ 𝑉 𝑑 (𝑧) ⇒ 𝜋(𝑥, 𝑠 𝑛 + 𝑡) ∈ 𝑉 𝑏 (𝜋(𝑥, 𝑡 𝑛 )) for all n all 𝑡 ∈ 𝑅. Thus we have two convergent nets {𝜋(𝑥, 𝑆 𝑛 + 𝑡)} 𝑎𝑛𝑑 {𝜋(𝑥, 𝑡 𝑛 + 𝑡)} in the compact hull H(x)=E, so they are uniformly convergent. Now for the index b ∈ A, there exists mo such that 𝜋(𝑥, 𝑆 𝑛 + 𝑡) ∈ 𝑉 𝑏 (𝑧, 𝑡)𝑎𝑛𝑑 𝜋(𝑥, 𝑡 𝑛 + 𝑡) ∈ 𝑉 𝑏 (𝑦, 𝑡) for all 𝑛 ≥ 𝑚 0 and 𝑡 ∈ 𝑅. Finally, for the uniformity condition (1) where 𝑥 1 = 𝜋(𝑦, 𝑡)𝑥 2 = 𝜋(𝑥, 𝑡 𝑛 + 𝑡), 𝑥 3 = 𝜋(𝑥, 𝑆 𝑛 + 𝑡) and 𝑥 4 = 𝜋 (z, t), it follows that π(y, t) ∈ V a (π(z, t). Conversely, let us assume that the flow π is equicontinuous on E. We show that E is almost periodic minimal. Since E is compact minimal, then by Birkhoff recurrence theorem every motion in E is recurrent. Therefore, for any 𝑥 ∈ 𝐸 and any index by the set E(b, π𝑥) = {𝜏: π(𝑥, 𝜏) ∈ 𝑉 𝑏 (𝑥)} is relatively dense in R. Let the index a be preassigned, then the resists 𝑏 ∈ 𝐴 for the equicontinuity of π. Thus we have. E(b, 𝜋(𝑥, 𝑡) ∈ 𝑉 𝑏 (𝑥)}𝑐. {𝑡: π(𝑥, 𝑡 + 𝜏) ∈ 𝑉 𝑏 (π(x, t)), for all t ∈ 𝑅} Assuring the almost periodicity of 𝜋𝑥. Therefore, E is almost periodic minimal.

1. Birkhoff GD. Dynamical Systems, revised editions, Amer. Math. Soc. Colloq. Publ., Amer Math. Soc., Providence, RI 1966, IX.
2. Gottschalk WH. Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc 1944;50:915-919.
3. Gottschalk WH. Almost periodic points with respect to transformation semi-groups, Ann. Math 1946;47:762766.