In this paper we discussed relation proximal points with many of  dynamical properties through studied topological transformation group , and it will given  necessary condition for proximal relation to be minimal set ,and introduce  new define replete set and semi-replete set by using concept of the replete set and semi-replete set as well as  we introduce that many of  new  relations  and theorem. Key words: Proximal point, replete proximal point, syndetic set, semi-replete set, minimal set, almost periodic point             .

Abdullah, Enas . Y

Alkiffai, Ameera N.

Hilal, Rana H.

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  100  Relation proximal point with some dynamical properties  Enas . Y Abdullah Ameera N.Alkiffai Rana H.Hilal Kufa University College  Of Education For Girls Kufa University College Of Education For Girls Kufa University College Of Education For Girls Department Of Mathematic Department Of Mathematic Department Of Mathematic    Abstract :      In this paper we discussed relation proximal points with many of  dynamical properties through studied topological transformation group , and it will given  necessary condition for proximal relation to be minimal set ,and introduce  new define replete set and semi-replete set by using concept of the replete set and semi-replete set as well as  we introduce that many of  new  relations  and theorem. Key words: Proximal point, replete proximal point, syndetic set, semi-replete set, minimal set, almost periodic point  . Introduction :      The first has been introduced the proximal point in topological transformation group (𝑍, 𝑇, 𝜋) with compact hausdorff 𝑍 was Ellis R. and Gottschalk W.[4][5].We show  relation proximal point with some dynamical properties (fixed point-minimal point-almost periodic ),also we studied replete proximal and semi-replete proximal independed on  replete set  and semi-replete set ,and it will be given a necessary and sufficient condition for proximal and semi-replete proximal to be  syndetic set ,and we studied the image of proximal points under epimomorphism ,we give some theorem about proximal points .We use symbol ∆ to indication the end. 1.Preliminaries:      In this section we given important concepts that we needed in this work.  Definition (1-1) [ 4]:      A topological group is  a set  𝑇 with  two structures:  1. 𝑇 is a group 2. 𝑇 is a topological space  Definition (1-2)[4]:       A subset 𝜌 of  T is said to be {𝑙𝑒𝑓𝑡}, {𝑟𝑖𝑔ℎ𝑡} syndetic in T if and only if  there exists a compact subset 𝑘 of  T such that 𝜌𝑘 = 𝑘𝜌 = 𝑇.  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  101  Definition (1-3) [4]    Let (𝑍, 𝑇) be topological group : 1. A subset 𝑤 of  T is said to be replete if for each compact set 𝛽 of  T there exist 𝑡1, 𝑡2 ∈ 𝑇 such that   𝑡1𝛽𝑡2 ⊂ 𝑤. 2. A subset 𝑤 of  T is said to be semi-replete if for each compact set 𝛽 of  T there exist 𝑡 ∈ 𝑇 such that   𝛽𝑡 ⊂ 𝑤.       Definition (1-4)[ 4]:       A right topological transformation group is a triple (𝑍, 𝑇, 𝜋) where 𝑍 is a topological space called the phase space , 𝑇 is a topological group called the phase group 𝜋: 𝑍 × 𝑇 →𝑍, 𝜋(𝑧, 𝑡) → 𝑥𝑡  is a continuous mapping such that :  1. 𝑧𝑒 = 𝑧 (𝑧 ∈ 𝑍),where e is the identity  of  𝑇 2. (𝑧𝑡)𝑏 = 𝑧(𝑡𝑏)   (𝑧 ∈ 𝑍, 𝑧, 𝑏 ∈ 𝑇)  Definition (1-5) [ 3]:       Let (𝑍, 𝑇, 𝜋)be a topological transformation group  1. 𝐴 subset  𝐴 ⊂Z is  said to be invariant set if  𝐴𝑇 = 𝐴. 2. 𝐴 non-empty closed invariant set  𝐴 ⊂Z is  said to be minimal set if  it contains no non-empty ,proper, closed invariant subset .     Definition (1-6)[ 4]             Let (𝑍, 𝑇, 𝜋)be a topological transformation group ,then the set 𝑧𝑇 = {𝑧𝑡: 𝑡 ∈ 𝑇}is called  the orbit of 𝑧 and the set 𝑧𝑇̅̅ ̅ the orbit closure of  𝑧.  Definition (1-7)      Let (𝑍, 𝑇, 𝜋) be a topological transformation group is said to be strongly effective if for each 𝑧 ∈ 𝑍, 𝑡 = 𝑒 there exist 𝑡 ∈ 𝑇 such that 𝑧𝑡 = 𝑧.  Definition (1-8) [5]        Let (𝑍, 𝑇, 𝜋) be a topological transformation group: 1. Let 𝑧 ∈ 𝑍 , is said to be Fixed point under 𝑇 if 𝑧𝑇 = 𝑧. 2. Let 𝑧 ∈ 𝑍 , is said to be minimal points if closure orbit is minimal set. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  102  3. Let 𝑧 ∈ 𝑍 , is said to be almost periodic points if for each invariant neighborhood  𝑉 of z there exist syndetic subset 𝐴 of  𝑇 such that 𝑧𝐴 ⊂ 𝑉 .  Definition (1-9) [ 5]      Let (𝑍, 𝑇, 𝜋) and (𝐻, 𝐺, 𝜎) be a topological transformation group, and 𝜏: 𝑍 → 𝐻 be continuous ,𝛾: 𝑇 → 𝐺 be continuous homomorphism then (𝜏, 𝛾): (𝑍, 𝑇, 𝜋) →  (𝐻, 𝐺, 𝜎)  is said to be homomorphism and ((𝑧, 𝑡)𝜋)𝜏 = ((z)𝜏, (𝑡)𝛾)𝜎.If  for  each 𝜏, 𝛾 onto then homomorphism is said to be epimorphism .  Theorem (1- 10):      Let (𝑍, 𝑇, 𝜋)be a topological transformation group, 𝑤𝑔 be replete subset of  𝑇  if  and only if  𝑤 is replete set. Proof: Let 𝑤𝑔 be replete subset of 𝑇 then for each compact subset 𝛽 of T there exist 𝑡1, 𝑡2 ∈ 𝑇 such that 𝑡1𝛽𝑡2 ⊂ 𝑤𝑔 , since 𝑇 group  there exist 𝑔−1 ∈ 𝑇 ∈ such that 𝑡1𝛽𝑡 = 𝑡1𝛽𝑡2𝑔−1 ⊂ 𝑤  for some 𝑡 ∈ 𝑇  thus 𝑤 is replete set. Same method  proof part other.  Remark (1- 11)      Let (𝑍, 𝑇, 𝜋)be a topological transformation group , the following statements are valid.  1. if  𝐾 compact subset of  T then 𝑔𝐾 compact set for some 𝑔 ∈ 𝑇. 2. if  𝐴 syndetic subset of  T then 𝑔𝐴 syndetic set for some 𝑔 ∈ 𝑇.  Theorem (2-12)      Let (𝑍, 𝑇, 𝜋) be a topological transformation group , (𝑛, 𝑚) ∈ 𝑃 , then (𝑍, 𝑇, 𝜋)    strongly effective. Proof: Let 𝑛 and 𝑚 are proximal points then for each index 𝜑 in 𝑍 there exist a 𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑  ,it is enough to show that 𝑡 = 𝑒,since 𝜑 be invariant then  (𝑛, 𝑚)𝑡 ∈ 𝜑𝑡 by hypothesis there exist 𝑡−1 ∈ 𝑇 ,it follows that (𝑛, 𝑚)𝑒 ∈ 𝜑, and (𝑛, 𝑚)𝑡 ∩ (𝑛, 𝑚)𝑒 ≠ ∅   this   lead   to  (𝑛, 𝑚)𝑡 ⊂ (𝑛, 𝑚)𝑒  and  (𝑛𝑡, 𝑚𝑡) = (𝑛𝑒, 𝑚𝑒). Clearly 𝑡 = 𝑒 therefore (𝑍, 𝑇, 𝜋)  strongly effective. ∆ 2.Main results       In this section , we introduce proximal point in topological transformation group and show that relation proximal point with dynamical properties  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  103  Definition ( 2-1) [1,2]      Let (𝑍, 𝑇, 𝜋)be a topological transformation group ,a two points n and m of  𝑍 are called proximal proved that for each index 𝜑 in 𝑍 there exist a 𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑.The set of all proximal pairs are called the proximal relation and are denoted by 𝑃(𝑍, 𝑇) or simply 𝑃.  Definition ( 2-2)      Let (𝑍, 𝑇, 𝜋)be a topological transformation group ,a two points n and m of  𝑍 are called replete proximal proved that for each index 𝜑 in 𝑍 there exist a a replete subset 𝑤 of T such that (𝑛, 𝑚)𝑤 ⊂ 𝜑.The set of all replete proximal pairs are called the replete proximal relation and are denoted by 𝑅𝑃(𝑍, 𝑇) or simply 𝑅𝑃.  Definition ( 2-3)      Let (𝑍, 𝑇, 𝜋)be a topological transformation group ,a two points n and m of  𝑍 are called semi-replete proximal proved that for each index 𝜑 in 𝑍 there exist a a semi-replete subset 𝑣 of T such that (𝑛, 𝑚)𝑣 ⊂ 𝜑.The set of all semi-replete proximal pairs are called the semi-replete proximal relation and are denoted by 𝑆𝑅𝑃(𝑍, 𝑇) or simply 𝑆𝑅𝑃.  Remark (2- 4)       Let (𝑍, 𝑇, 𝜋)be a topological transformation group then: 1. P(X,T) is invariant set. 2. P(X,T) is close set .  Theorem (2-5)       Let (𝑍, 𝑇, 𝜋) be a topological transformation abelain group,(𝑛, 𝑚) are replete proximal points then 𝜑 is invariant . Proof: We may assume that T be replete group ,let 𝑛 and m are replete proximal points then for each index 𝜑 in 𝑍 there exist a replete subset 𝑤 of T such that (𝑛, 𝑚)𝑤 ⊂ 𝜑 and (𝑛, 𝑚)𝑤𝑇 ⊂ 𝜑𝑇 so (𝑛, 𝑚)𝑤𝑇 ∩ (𝑛, 𝑚)𝑤 ≠ ∅  thus 𝜑𝑇 ⊂ (𝑛, 𝑚)𝑤 ⊂ 𝜑 since 𝜑𝑒 ⊂ 𝜑𝑇 ,thus  𝜑 is invariant under T. Theorem (2- 6)      Let (𝑍, 𝑇, 𝜋)be a topological transformation abelain group,(𝑛, 𝑚) are proximal points if and only if (𝑛, 𝑚) are replete proximal points. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  104  Proof: Suppose that 𝛽 be compact subset of T , since 𝑛 and 𝑚 are proximal points then for each index 𝜑 in 𝑍 there exist a 𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑  , by hypothesis there exist identity 𝑒 in T such that 𝛽𝑒 ⊂ 𝑇 so 𝛽𝑔𝑔−1 ⊂ 𝑇 for all 𝑔 ∈ 𝑇 , 𝛽𝑔 ⊂ 𝑇𝑔 ⊂ 𝑇 and 𝑔1𝛽𝑔 ⊂𝑔1𝑇 ⊂ 𝑇 for all 𝑔1 ∈ 𝑇 .Therefore 𝑇 be replete  and 𝑛 and 𝑚 are replete proximal points. Conversely let 𝑛 and 𝑚 are replete proximal points ,then for each index 𝜑 in 𝑍 there exist a replete subset 𝑤 of T such that (𝑛, 𝑚)𝑤 ⊂ 𝜑, since 𝑤 replete subset of T then for each compact set 𝛽 of T there exist 𝑡1, 𝑡2 ∈ 𝑇 such that 𝑡1𝛽𝑡2 ⊂ 𝑤 , (𝑛, 𝑚)𝑡1𝛽𝑡2 ⊂ (𝑛, 𝑚)𝑤 ⊂ 𝜑 so (𝑛, 𝑚)𝑡1𝛽𝑡2𝑇 ⊂ 𝜑𝑇 by theorem (2-5) we obtain (𝑛, 𝑚)𝑡1𝛽𝑡2𝑇 ⊂ 𝜑 by hypothesis (𝑛, 𝑚)𝛽𝑡1𝑡2𝑇 ∩ (𝑛, 𝑚)𝛽𝑇 ≠ ∅, 𝑡ℎ𝑢𝑠 (𝑛, 𝑚)𝛽𝑇 ⊂ 𝜑 since 𝑇 syndetic set and 𝛽 compact set then (𝑛, 𝑚)𝑇 ⊂ 𝜑 . ∆  Theorem (2-7)      Let (𝑍, 𝑇, 𝜋) be a topological transformation group,(𝑛, 𝑚) are proximal points uf and only if  (𝑛, 𝑚) are semi-replete proximal points. Proof: Suppose that 𝜇 be compact subset of T , since 𝑛 and 𝑚 are proximal points then for each index 𝜑 in 𝑍 there exist a 𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑  ,we need to prove T be semi-replete , by hypothesis 𝜇𝑡 ⊂ 𝑇𝑡 ⊂ 𝑇 .Then T be semi-replete proximal thus and 𝑛 and 𝑚 are proximal points. Conversely let 𝑛 and 𝑚 are semi-replete proximal points ,then for each index 𝜑 in 𝑍 there exist a semi- replete subset 𝑤 of T such that (𝑛, 𝑚)𝑤 ⊂ 𝜑, since 𝑤 semi-replete subset of T then for each compact set 𝜇 of T there exist 𝑡1 ∈ 𝑇 such that 𝑡1𝜇 ⊂ 𝑤 ,and  (𝑛, 𝑚)𝑡1𝜇 ⊂ 𝜑 so (𝑛, 𝑚)𝑡1𝜇𝑇 ⊂ 𝜑𝑇 by theorem ( 2-5) we obtain (𝑛, 𝑚)𝑡1𝜇𝑇 ⊂ 𝜑 by hypothesis  (𝑛, 𝑚)𝜇𝑇 ⊂ 𝜑 since 𝑇 syndetic set and 𝛽 compact set then (𝑛, 𝑚)𝑇 ⊂𝜑  (𝑛, 𝑚) are proximal points . ∆  Theorem (2-8)      Let (𝑍, 𝑇, 𝜋)  be a topological transformation abelain group,(𝑛, 𝑚) are replete proximal points if and only if (𝑛, 𝑚) are semi replete proximal points. Proof: let 𝑛 and 𝑚 are replete proximal points ,then for each index 𝜑 in 𝑍 there exist a replete subset 𝑤 of T such that (𝑛, 𝑚)𝑤 ⊂ 𝜑, since 𝑤 replete subset of T then for each compact set 𝛽 of T there exist 𝑡1, 𝑡2 ∈ 𝑇 such that 𝑡1𝛽𝑡2 ⊂ 𝑤 , (𝑛, 𝑚)𝑡1𝛽𝑡2 ⊂ (𝑛, 𝑚)𝑤 ⊂ 𝜑 by hypothesis we get 𝛽𝑡 ⊂ 𝑤, for some 𝑡 ∈ T then 𝑤 be semi- replete proximal subset of T thus(𝑛, 𝑚) ∈𝑆𝑅𝑃 . Conversely let 𝑛 and 𝑚 are semi-replete proximal points ,then for each index 𝜑 in 𝑍 there exist a semi- replete subset 𝑤 of T such that (𝑛, 𝑚)𝑤 ⊂ 𝜑, since 𝑤 semi-replete subset of T then for each compact set 𝜇 of T there exist 𝑡1 ∈ 𝑇 such that 𝑡1𝜇 ⊂ 𝑤 so 𝑡1𝜇𝑒 =𝑡1𝜇 𝑡. 𝑡−1 ⊂ 𝑤  for some 𝑡−1 ∈ 𝑇 and 𝑡1𝜇 𝑡 ⊂ 𝑤𝑡 then 𝑤𝑡 replete subset of T from theorem (1-10) we get 𝑤 be replete set and (𝑛, 𝑚) ∈ 𝑅𝑃. ∆   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  105  Theorem (2- 9)       Let (𝑍, 𝑇, 𝜋) be a topological transformation group,(𝑛, 𝑚) are proximal points then  (𝑍, 𝑇) are pointwise proximal points.  Proof: Let (𝑛, 𝑚) ∈ 𝑃 ,it follows that  (𝑛, 𝑚)𝑇 ⊂ 𝑃𝑇  clearly by remark (2-4) that  (𝑛, 𝑚)𝑇 ⊂𝑃  so (𝑛, 𝑚)𝑇𝑇 ⊂ 𝑃𝑇, (𝑛, 𝑚)𝑇2 ⊂ 𝑃 again (𝑛, 𝑚)𝑇2𝑇 ⊂ 𝑃𝑇, (𝑛, 𝑚)𝑇3 ⊂ 𝑃 after n- time we get  (𝑛, 𝑚)𝑇𝑛 ⊂ 𝑃 then we obtain that  (𝑍 × 𝑍, 𝑇) are proximal  for each point in 𝑍 × 𝑍 therefore (𝑍, 𝑇) are pointwise proximal points. ∆   Theorem (2-10)       Let (𝑍, 𝑇, 𝜋) be a topological transformation abelain group , 𝐹 ⊂ 𝑇 𝑎𝑛𝑑 (𝑛, 𝑚) ∈ 𝑃  𝑡ℎ𝑒𝑛 𝐹 syndetic set . Proof: Let 𝐹 be subset  of  𝑇  ,since (𝑛, 𝑚) are proximal points then for each index 𝜑 in 𝑍 there exist  𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑 by theorem  (2-5) we have 𝜑𝑇 = 𝜑 and 𝜑𝑇𝐹 = 𝜑𝐹 since 𝑇 be syndetic there exist compact set 𝛽 of T  such that 𝜑𝑇𝐹𝛽 = 𝜑𝐹 𝛽 since  𝑇 𝑏𝑒 𝑎𝑏𝑒𝑙𝑖𝑛𝑒  then  𝜑𝑇𝐹 ∩  𝜑𝐹𝛽 ⊂ 𝜑𝑇 ∩  𝜑𝐹𝛽  and 𝜑𝑇𝐹 ∩  𝜑𝐹𝛽 ≠ ∅,Thus 𝜑𝑇 ∩  𝜑𝐹𝛽 ≠ ∅ so 𝑇 ⊂ 𝐹𝛽 by hypothesis  𝑇 = 𝐹𝛽 then 𝐹 syndetic set.∆  Theorem (2-11)       Let (𝑍, 𝑇, 𝜋) be a topological transformation abelain group , 𝐹 ⊂ 𝑇 𝑎𝑛𝑑 (𝑛, 𝑚) ∈ 𝑅𝑃  𝑡ℎ𝑒𝑛 𝐹 syndetic set . Proof: Let (𝑛, 𝑚) are replete proximal points then for each index 𝜑 in 𝑍 there exist replete subset E of  𝑇 such that (𝑛, 𝑚)𝐸 ⊂ 𝜑 since 𝐸 replete set then for each compact set  𝐾 in T there exist 𝑔1𝑔2 ∈ 𝑇 such that 𝑔1𝐾𝑔2 ⊂ 𝐸 since T by syndetic and 𝐾 by remark (1-11 number (1)) we have   𝑇𝑔2 ⊂ 𝐸  and  𝑇𝑔2𝐾 ⊂ 𝐸𝐾 so 𝑇 ⊂ 𝐸𝐾 by hypothesis  we obtain 𝑇 = 𝐸𝐾 therefore 𝐸 be syndetic set. ∆  Remark (2-12)       Let (𝑍, 𝑇, 𝜋)  be a topological transformation abelain group , 𝐹 ⊂ 𝑇 , 𝑖𝑓(𝑛, 𝑚) ∈ 𝑆𝑅𝑃  𝑡ℎ𝑒𝑛 𝐹 syndetic set . Theorem (2-13)       Let (𝑍, 𝑇, 𝜋) be a topological transformation group then P(X,T) is minimal set. Proof: Let (𝑛, 𝑚) ∈ 𝑃 and (𝑛, 𝑚)𝑇 ⊂ 𝑃𝑇 by remark (2-4 number (1)) we obtain (𝑛, 𝑚)𝑇 ∈ 𝑃 , so (𝑛, 𝑚)𝑇 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅  ⊂  ?̅? then  (𝑛, 𝑚)𝑇 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅  ⊂ 𝑃 by remark (2-4 number (2)) since (𝑛, 𝑚)𝑇̅̅ ̅̅ ̅̅ ̅̅ ̅̅  be least Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  106  closed invariant  subset 𝑍 ×Z contain (𝑛, 𝑚) therefore  𝑃 ⊂ (𝑛, 𝑚)𝑇 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ ,thus 𝑃 ⊂ (𝑛, 𝑚)𝑇 ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ then 𝑃 be minimal set and (𝑛, 𝑚) be minimal points . ∆  Theorem (2-14)      Let (𝑍, 𝑇, 𝜋)be a topological transformation group then (𝑛, 𝑚)is fixed points.  Theorem (2-15)       Let (𝑍, 𝑇, 𝜋) be a topological transformation group , (𝑛, 𝑚) ∈ 𝑃 , if and only if (𝑛, 𝑚) is almost periodic points. Proof: Assume that 𝑉 be invariant neighborhood of (𝑛, 𝑚) and 𝐴 subset of  𝑇 It is enough to show that 𝐴  be syndetic set ,since (𝑛, 𝑚) are proximal points then for each index 𝜑 in 𝑍 there exist  𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑 and (𝑛, 𝑚)𝐴 ⊂ (𝑛, 𝑚)𝑇 ⊂ 𝜑. It follows that 𝐴 be syndetic set by hypothesis (𝑛, 𝑚)𝐴 ⊂ 𝑉𝐴 ⊂ 𝑉𝑇 ⊂ 𝑉  ,thus (𝑛, 𝑚) is almost periodic points .Conversely assume that (𝑛, 𝑚) ∈ 𝜑 since (𝑛, 𝑚) almost periodic point then for each invariant neighborhood  𝑉 of (𝑛, 𝑚) there exist syndetic subset 𝐴 of  𝑇 such that (𝑛, 𝑚)𝐴 ⊂ 𝑉  , (𝑛, 𝑚)𝐴𝑔 ⊂ 𝑉𝑔 = 𝑉 for some 𝑔 ∈ T .It is enough to show that 𝐴𝑔 be semi- replete set ,since 𝐴 be syndetic set then 𝐴𝑔 be syndetic set by remark (1-11 number (2))there exist compact subset 𝐾 of  𝑇 such that 𝐴𝑔𝐾 = 𝑇  ,for each 𝑡 ∈ 𝑇 there exist 𝑎 ∈ 𝐴, 𝑘 ∈ 𝐾 such that 𝑎𝑔𝑘 = t so 𝑎𝑔𝑘 = 𝑡𝑘−1𝑔−1 for some 𝑘−1, 𝑔−1 ∈ 𝑇 ,it follows that 𝑡𝐾 ⊂ 𝐴𝑔 thus 𝐴𝑔 be semi-replete set by hypothesis it was found (𝑛, 𝑚)𝐴𝑔 ⊂  𝜑, then (𝑛, 𝑚) be semi-replete proximal points by theorem (2-7) we obtain (𝑛, 𝑚) ∈ 𝑃. ∆  Theorem (2-16)      Let(𝜏, 𝛾): (𝑍, 𝑇, 𝜋) →  (𝐻, 𝐺, 𝜎) epimorphism , (𝑛, 𝑚) are proximal points under 𝑇 then ((𝑛)𝜏, (𝑚)𝜏) are proximal points under 𝐺. Proof: Let (𝑛, 𝑚) are proximal points then for each index 𝜑 in 𝑍 there exist  𝑡 ∈ 𝑇 such that (𝑛, 𝑚)𝑡 ∈ 𝜑.It follows that from define homomorphism we obtain   ((𝑛)𝜏, (𝑚)𝜏)𝑇𝛾 ∈ (𝜑)𝜏 , since 𝛾 be onto then we get ((𝑛)𝜏, (𝑚)𝜏)𝐺 ∈ (𝜑)𝜏 therefore  ((𝑛)𝜏, (𝑚)𝜏)𝐺 ⊂ (𝜑)𝜏 thus ((𝑛)𝜏, (𝑚)𝜏) are proximal points under 𝐺. ∆ .    Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.8, 2016  107          References  1. Auslander .J, " On the proximal relation in topological dynamics " Proc. Amer. Math .Soc. II (1960) ,88- 86.  2. Clay. J, " Proximity relations in transformation groups, Trans. Amer. Math. Soc .108(1963),88-96.  3. Dikranjan .D,  and  Megrelishvili. M ," Minimality conditions in topological groups in :recent progress in general topology , springer ,atlantis  press ,III(2014) ,229-327,   4. Ellis. R ," Lectures on topological dynamics " ,W.A. Benjamin ,New York, 1969.  5. Ellis. R and Gottschalk .W ," Homomorphism of transformation groups" Trans. Amer .Math .Soc .94(1960),258-271.   6. Keynes. H , " On the proximal relation being closed "  Proc. Amer. Soc.81(1967),518-522.     Proximal point Semi-Replete proximal point Replete proximal point Minimal point Fixed point Almost periodic point 

Relation proximal point with some dynamical properties

https://www.iiste.org/Journals/index.php/MTM/article/download/32173/33060

Relation proximal point with some dynamical properties

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