Proximal Point in Topological Transformation Group النقطت القريبت في زمرة التحويل التبولوجي

Transcription

1 Proximal Point in Topological Transformation Group النقطت القريبت في زمرة التحويل التبولوجي ايىاس يحي عبد هللا Abstract In this paper, we introduce the concept of proximal points in topological transformation group and it will be given a necessary and sufficient condition for a proximal points to be replete proximal points and almost periodic points also we show relation proximal points by syndetic set in side and locally almost periodic in other side الخالصت في هذا البحث قدمىا مفهىم الىقطت األقزب في سمزة التحىيل التبىلىجيت كما سىف وعطي الشزوط الضزوريت والكافيت لتكىن الىقطت األقزب وقطت اقزب مفعمت ووقطت دوريت تقزيبا كذلك بيىىا عالقت الىقطت األقزب بالمجمىعت الزابطت مه جهت و بالدوريت تقزيبا محليت مه جهت أخزي Introduction The proximal relation P( in transformation group ( with compact hausdorff X has been introduced by R Ellis and WH Gottschak (12) and studied in (3), ( 4 (,(5) The proximal relation is a reflexive, symmetric The proximal relation plays an important role to characterize the transformation groups The notions of replete proximal relation and replete regular relation has been strengthened and extended by consideration of replete set In (6), the author introduced the r- homomorphism between two transformation groups The author(7) introduced regular relation R( on basis of the proximal relation P( This paper investigates the properties of there proximal relations and show some relations between proximal and replete proximal We also give some necessary and sufficient conditions for the regular relation and proximal relation to be regular We studied inverse image proximal points 2Preliminaries Throughout this paper,( will denote a transformation group with compact hausdorff phasa spase A closed invariant non-empty subset A of X is called minimal set if dose not have any proper subset with these three properties A point whose orbit closure is minimal set is called almost periodic point, ( X, transformation group is said to be free effective if there exist x X with xt then t e for each t T A subset A of T is said to be {left}{right} syndetic in T provided that for some compact subset K of T AK T KA and is said to be replete if for each compact set K there exist g, g 1 such that gkg 1 AA continuous map : ( ( with xt ) t is called a homomorphism if Y is minimal, is always onto Especially, if is onto, is called an epimorphism A homomorphism from ( T ) onto itself is called an endomorphism of (, and an isomorphism : ( ( is called an automorphism of ( A homomorphism : ( ( is called distal if (, x y 132

2 Definition (2-1)( 1 2)Let ( X, be a transformation group a two points x and y of X are called proximal proved that for each index in X,there exist a t T such that ( xt, The set of all proximal pairs is called the proximal relation and is denoted by P( X, or simply P Definition (2-2)Let ( X, be a transformation group two points x and y of X are called replete proximal proved that for each index in X,there exist a replete subset A T such that ( xt, The set of all replete proximal pairs is called the replete proximal relation and is denoted by RP ( X, or simply RP Definition (2-3) (8)A transformation group ( X, is called locally almost periodic if for each neighborhood U of x there is a neighborhood V of x and a syndetic subset A T withva U ( X, is said to be locally almost periodic if it is locally almost periodic at each point x X Definition (2-4) A minimal transformation group ( X, is called regular minimal if ( x, is an almost point of ( X and, there is an automomorphism h of ( X, such that h( y Definition (2-5) ( 5 ) The homomorphism : ( ( is said to be regular if for each y 1 (, there exists an automomorphism h of ( X, such that ( h and h (y Y) Theorem (2-6) Let ( X, be transformation group the fallowing hold: 1- If Then P ( X, be invariant set 2- If Then P ( X, be closed set 3- and ( almost periodic Then P ( X, be minimal set 4- If then ( h (, T Proof: (3): Let P, since ( x, T be least closed invariant subset of X X contain ( x, such that P T, since almost periodic then ( x, T be a minimal set by theorem ( 2-6 (1)(2)) we have obtain P ( X, be minimal set Theorem (2-7) Let ( X, be Transformation group and P Then the following hold: 1) ( x, T be minimal set 2) If X X be a minimal set then P be dance set Proof: We need only prove (1)Let P,then T PT and from theorem (2-6 number (1))we obtain T P,so ( x, T P and P is closed set by theorem (2-6 number (2) ),Then ( x, T P,since ( x, T be least closed invariant subset of X X contain therefore P T Thus we have P T therefore ( x, T be minimal set Theorem (2-8):Let ( X, be transformation group and P Then is invariant 133

3 Proof: Let A be replete subset of T and P we obtain ( x, A by hypothesis there exists t T such that T for all index of X and TT since T be an invariant we have T ( x, T A T A and ( x, T A Thus A so A by hypotheses we obtain then therefore is invariant Theorem (2-9) Let ( X, be Transformation group free effective and P If and only if RP P, then for each index of X there exist t Such that ( x, T Since T be replete we obtain RP Conversely Let RP then for each index of X there exist A replete subset of T such that ( x, A,since A replete subset of Tthen for each compact set K of T there exist g 1, g 2 such that g 1 Kg 1 A and ( x, g1kg2 and g1kg 2T by theorem (2-8) we get ( xg1, yg1) Kg2T since ( X, be free effective then Kg 2T and Kg be compact set by hypothesis we have ( x, T therefore P 2 Theorem (2-10): If ( X, be locally almost periodic and U X then fallowing hold: 1) U be invariant set 2) If U be neighborhood of y then U be neighborhood of x Proof: We need only prove (1) Let ( X, be locally almost periodic then for each neighborhood U of x there exist neighborhood V of x and syndetic subset A of T such that VA U -1 VAK UK, VT UK Then for each u U there exist t T such that xt uk, xtk u so that xt U, Since U neighborhood of x and xt orbit of x Then U xt thus we have U xt therefore U be invariant set Theorem (2-11) Let ( X, be Transformation group, P and U be neighborhood of x then ( X, be locally almost periodic Proof: Suppose that V be neighborhood of x,and P In fact take h to be the identity therefore there exist syndetic set A of T such that Ixa ya for some a A,by hypotheses we have xa xa VA Since U be neighborhood of x then xa UA UT by theorem (2-10(1)) we get xa U and AV U and AV U Therefore ( X, be locally almost periodic Theorem (2-12):Let ( X, be transformation group, T is abelain and P, then A be syndetic set Proof: Let A be subset of T we show that A be syndetic set, since P, Then for each index in X there existt T, such that ( x, T by theorem (2-8) we have and A A since T be syndetic set there exist compact set K of T such that AK AK and T be abelain group implies KA AK and A AK,by hypothesis, A,thus AK T AK Therefore A be syndetic set 134

4 Theorem (2-13):If : X Y be a distal homomorphism ( X, be regular minimal if and only if be regular Proof: Let Since ( X, be regular minimal and almost periodic point of ( X,then there exist an outomomorphim h in ( X, such that h( y and therefore we have ( h(,, since be distal and h ( ( We have h Thus we obtain is regular Conversely suppose that be almost periodic point in ( X,we need to prove ( X, be regular minimal,because regular there exist automomorphism h of ( such that h( t yt for some t T,by hypotheses there exist t -1 T such that 1 h( tt ytt implies h y ( therefore ( X, be regular minimal Remark (2-14):Let ( X, be regular and : X Y be homomorphism then be distal Remark (2-15): 1 If ( be regular then P 2 If ( X, be regular minimal then ( X, be locally almost period Theorem ( 2-16 ): If x and y be replete proximal then RP be invariant set RP,then for each index in X,there exist a replete subset A T such that ( xt, since A be replete subset then for each compact sub set there exist g 1, g 2 such that g 1 Kg 2 A, that is g1kg2 and g1kg 2T by(2-8)we have g1kt since T syndetic set and g 1 K compact set Then ( x, T and so P therefore RP P by theorem (2-9)Thus we have RP P and RP be invariant set Remark ( 2-17): If P ( X, closed set and X be compact then P be compact set Theorem (2-18):Let ( X, be transformation group and RP A is replete, Then A is sub group Proof: Let A a A g kg a : 1 2 is replete set and RP Then for each K compact subset of T there exist g 1, g 2 Such that g 1 Kg 2 A,Let a, b A and g1kg2 a, g1kg2 b for all k K by hybothsis, there exist b with ab g kg b, since T be closed then g2 b In fact take g2 b d thus we get g kd ab, therefore ab A,and A is replete group 1 Theorem (2-19):Let ( X, be transformation group 3-A Characterization of regular In this section, we give a characterization of regular Definition (3-1) (7)Let ( X, be a transformation group The points x and y of X are said to be regular if h( and y are proximal for some automorphism h of (The set of all regular pairs is called the regular relation and is denoted by R ( X, or simply R Definition (3-2) Let ( X, be a transformation group Two points x and y are to replete regular if h( and y are replete proximal for some automorphism h of X The set of all regular pairs is called the regular relation and is denoted by R ( X, or simply R 135

5 Definition (3-3)(9)Let ( X, and ( be transformation groups An epimorphism : X Y is called an r-homomorphism if a given h in ( X,,there exists a k in ( Y, such that h k Theorem (3-4) If R, and : X Y be a r-homomorphism and :T T be onto homomorphism then ( (, ) Proof: Let We need show that ( (, ),since R,Then there exist automorphism h of ( X, such that ( h P Therefore ( h t yt (t, since : X Y is r-homomorphism then there exist K in ( such that h k Therefore we have k ( t) k ( xt) h( xt) h( t) t) Huns we have ( k (, ) and therefore ( (, ) Theorem (3-5)Let : ( ( be r-homomorphism distal ( X, be regular minimal and then ( (, ( ) Proof: Suppose that We need only to show that ( (, ( ) Since ( X, is regular minimal and is an almost periodic point of ( X,there exists an automomorphism k in ( such that k( y and k( ( since be r- homomorphism there exist an 1 automomorphism h in ( such that h k is equivalent h k implies h( ( ) ( we have ( h( ( ), ( and is distal then we get ( (, ( Definition (3-6)Let f : X g : X X Then f and g are conjugate if there is a homomorphism h : X X such that h g f h In this case, we write f g Theorem (3-7):Let ( X, be transformation group and R f : X g : X X If x be fixed under g and y fixed under f then f h g R, then there exist automomorphism h in ( Such that h( y for some e T and f h( f ( h( ) f ( by hypothesis we have f h( y so h g( h( g( ) h( y therefore f h( h g( Theorem (3-8):Let ( X, be transformation group, f : X X bijective h( fixed under f Then R h Theorem (3-9): Let ( X, be Transformation group free effective If then R R ( R R then there exist automomorphism h of ( Then ( h RP from theorem (2-9) we have ( h P,then R therefore R R similar method we have R R R R therefore Remark (3-10): Let : X Y be a distal homomorphism ( X, be regular minimal then 136

6 Theorem (3-11)Let : X Y be homomorphism, A be syndetic set and R, then fallowing hold: 1) If y be fixed pointthen there exist an idempotent b in A 2) A is subgroup of T Proof: (1)Let R Then there exist automomorphism h in ( Such that h( a ya for some a A and A is syndetic set now have h(xa) (h(a) ( ya) since y be fixed point then h(xa) h( Now have xa x therefore a e Thus A has contain an idempotent (2) direct proof from (1) Theorem (3-12):If : X Y be a homomorphism Then following hold 1) If then R is an invariant set 2) If then R is an closed set Theorem (3-13):If : X Y be a distal homomorphism and be regular if and only if R,then there exist automorphism h of ( X, such that ( h P From theorem (2-6(4)) we have ( hxt, P for all t, that is ( xt, R and T R by hypotheses we obtain R T,therefore R T,from theorem (3-11(1),(2)) we get R T and ( x, T be minimal set then ( x, be almost periodic by hypotheses there exist e T such that h( y and h( since be distal then h( thus we get h Therefore be regular Conversely direct from the definition of regular References (1) Ellis R Lectures on Topological Dynamics, W A Benjamin, New York, (2) Ellis R and Gottschalk W H, Homomorphisms of transformation groups, Trans Amer Math Soc 94 (1960), (3) Auslander J, On the proximal relation in topological dynamics, Proc Amer Math Soc (1960), (4) Clay J P,Proximity relations in transformation groups, Trans Amer Math Soc 108 (1963), (5)WooMH,Distal homomorphism,j of KMS,14(1978), (6)YuJO and Shin SS,The relative regionally regular relation,jchungcheong Math,Soc20(2007),no4, (7) Yu J O Regular relations in transformation group, J Korean Math Soc 21 (1984), (8) Keynes HBOn the proximal relation being closed,proc Amer,Soc18(1967,

7 138