New Characterization of Topological Transitivity

Transcription

1 ew Characterizatio o Topological Trasitivity Hussei J Abdul Hussei Departmet o Mathematics ad Computer Applicatios, College o Sciece, Uiversity o Al Muthaa, Al Muthaa, Iraq Abstract Let be a dyamical system, is said to be topological trasitive i or every pair o o-empty ope set, there exists such that We itroduce ad ivestigate a ew deiitio o topological trasitive by usig the atio -ope subset ad we called -trasitive ad prove the equivalet deiitios o this ew deiitio Keyword: Topological trasitive,-ope Subset MSC:54H2 وصف جديد للنتقالية التبولوجية حسين جابر عبد الحسين قسم الرياضيات وتطبيقات الحاسوب,كلية العلوم, جامعة المثنى,المثنى,الع ارق الخلصة قدم البحث وصفا جديدا للدوال االنتقالية وباستخدام مفهوم المجموعة (-Ope) وسمية هذا المفهوم االنتقالية )-trasitive(, وتم تقديم عدد من التعريفات المكافئة لهذا المفهوم الجديد 1Itroductio The cocept o topological trasitive goes back to GD Birkho who itroduced it i 192, [1] A topological trasitive dyamical system has poits which evetually move uder rom oe arbitrarily small ope set to ay other Cosequetly, such a dyamical system caot be decomposed ito two disjoit sets with oempty iteriors which do ot iteract uder the trasormatio We will cosider a discrete dyamical system give by a metric space ad cotiuous map The trajectory o a poit beig the sequece,where is the th iteratio o The set o poits o the trajectory o uder is called the orbit o,deoted by A poit is called o-waderig i or every eighborhood o there is a positive iteger such that [2] The set o o-waderig poits o will be deoted by Hussei_almaaly77@yahoocom 817

2 A dyamical system is said to be topological trasitive (hereater briely called trasitivity),i or every pair o o-empty ope set, there exists such that, [3] Recetly several researches were coducted to itroduce weak orms o ope set ad obtai some characterizatio ad preservig theorem o topological properties Al Omari A ad oorai M i [4] itroduce ew class o set called - ope sets "A subset o a space is said to be a -ope i or every there exists a ope subset i cotais such that is a iite set" They prove that the amily o all -ope establishes a topology Moreover,they obtai a characterizatio ad preservig theorem o compact space The objective o this paper is use the ew class o -ope set to create a - trasitive uctio ad prove some equivalet deiitios 2 -Ope Set I 29 Al Omari ad oorai[4], itroduce the cocept o "-ope Set" i a topological space, they prove that the amily o all -ope sets establishes a topology Moreover, they obtai a characterizatio ad preservig theorems o compact spaces Deiitio21[4]: A subset o a space is said to be a - ope i or every there exists a ope subset i cotais such that is a iite set The complemet o -ope set is said to be a -closed set Clearly every ope set is -ope but the coverse is ot true I is a topological space the the amily o all -ope subsets o is a topological space, [4] Theorem22[4]: Let be the topological space the the amily o all -ope subset is a topological space The uio o all -ope sets o cotaied i a subset is called the -iterior o ad deoted by [5] Propositio 23 [5]: A subset o a space is -ope i ad oly i Clearly, the iterior o is a subset o ( The -eighborhood o a poit is ay -ope subset o which cotais ow we ca itroduce the ollowig deiitio Deiitio 24: A subset is said to be -owhere dese i the closure has empty -iterior The uctio is said to be cotiuous uctio i is a ope set i or every ope set i This deiitio is equivalet to the ollowig deiitio see[4], a uctio is said to be -cotiuous i is a - ope set i or every ope set i, i ad is -ope the is said to be *- cotiuous, [5],clearly every cotiuous is a -cotiuous uctio, but the coverse is ot true i geeral, ad every *- cotiuous uctio is a -cotiuous,but the coverse is ot true i geeral, or more details see [5] The ollowig propositio gives the coditio o a cotiuous uctio which implies *- cotiuous Propositio 25 [5]: I be cotiuous ijective,the is -ope wheever is a -ope subset o A subset o topological space is said to be dese set (or everywhere dese ) i,i the closure o is equal to [6] Equivaletly, is dese i ad oly i itersects every o-empty ope set i ow,we ca prove the relatio betwee dese set ad -ope set i a topological space Theorem 26: let be a topological space ad the is dese i i ad oly i or every o-empty -ope set 818

3 let be a dese subset o,the or every o-empty ope set, Let be -ope subset i, suppose implies,but is -ope the there exist is ope set i such that,ad is iite this is a cotradictio, Coversely, suppose or every oempty -ope set o, we shall show that is dese i ot suppose ad, so, which is a cotradictio to that act is ope set i,so is -ope set i Hece, thereore the, ( is dese) 3 -Trasitive Map I this sectio we itroduce the ollowig ew otio Deiitio 31 let be a compact metric space ad a cotiuous map The map is said to be -trasitive i or all o empty -ope sets there exists such that Clearly every trasitive map is -trasitive but the coverse is ot true The ollowig results gives a equivalet coditio or -trasitivity Theorem 32: Let ( X, ) be a dyamical system,the is topologically - trasitive i ad oly i or every o-empty -ope set U i X, ( U ) is dese i X Proo : Assume ( U ) is ot dese The there exists a o-empty -ope set V such that ( U) V This implies ( U) V or all This is a cotradictio to the -trasitivity o Hece ( U ) is dese i X ow, let U adv be two o-empty -opes sets i X Theorem 26, we have ( U ) is dese i X so, by ( U) V This implies there exists m such that m ( U) V, hece is -trasitive ow, i is dyamical system ad is *-cotiuous the the ollowig coditio are equivalet Theorem33: let be a dyamical system ad is *- cotiuous the the ollowig are equivalet: (i) is -trasitive (ii) For every o-empty -ope set U i X, ( U ) is dese i X (iii)i E X is -closed ad ( E) E the E X or E is -owhere dese 1 (iv) I U X is -ope ad ( U) U the U or U is dese i X (i)(ii) Sice is *-cotiuous ad the amily o -ope set is topological space,thus ( U ) is -ope ad sice is - trasitive, it has to meet every -ope set i X ad hece is dese (ii) (i) Let U, V be -ope ad o-empty sets i X The As a result ( U ) is dese i X U ( V ) This implies m m such that U (V ) We urther have m ( U m m ( V)) ( U) V Hece is -trasitive (i) (iii) is -trasitive, E X is - closed ad ( E) E Assume that E X 819

4 ad E has a o-empty -iterior Deie U X \ E ClearlyU is -ope, sice E is -closed Let V E be -ope sice E has a o-empty -iterior We have ( V ) E sice E is ivariat The ( V ) U or all This is a cotradictio to - trasitivity Hece E X or E is -owhere dese (iii) (i) LetU be o-empty -ope set i X Suppose is ot -trasitive,the rom(ii) o this theorem, dese,but ( U ( U ) is ot ) is -ope Deie E X \ ( U) Clearly E is -closed ad E X Claim ( E) E Suppose (E) is ot a subset o E This implies ( E) ( U ) This urther implies 1 ( E) ( U) E ( U) o This is cotradictio to the deiitio o E, thus ( E) E Sice ( U ) is ot dese, there exists a o-empty -ope set V i X such that ( U) V This implies V E,this is cotradictio to the act that E is -owhere dese Hece is -trasitive (i)(iv) is -trasitive, U X is -ope 1 ad ( U) U Assume that U ad U is ot dese i X The there exists a oempty -ope set V i X such that U V Further ( U) V or all This implies U (V ) or all,a cotradictio to -trasitivity o Hece U or U is dese i X (iv) (i) Suppose is ot -trasitive, or a o empty -ope set U i X, let W ( U) is o-empty,-ope ad ot 1 dese Clearly ( W) W, this is cotradictio sice W is dese This proves that is -trasitive We ca itroduce the ew deiitio o owaderig poit Deiitio34: A poit is called - o-waderig i or every -eighborhood o there is a positive iteger such that The set o all -owaderig poits o will be deoted by Topological trasitivity ad existece o a dese orbit are two equivalet deiitio or some space but is ot true geerally, [7, 3]I the ollowig we will make a coectio betwee the set o -o waderig poits, -trasitive ad a dese orbit Propositio35: Let : X X be a cotiuous map o compact metric space, is -trasitive i ad oly i ( ) X,ad has a dese orbit Suppose is - trasitive, clearly has a dese orbit, ie,there exits,such that is dese i, i ( ) X the there exist a o-empty -ope subset such that are pairwise disjoit set Sice is dese orbit, or some,,,which cotradictio with is pairwise disjoit set Thereore ( ) X ow, suppose has a dese orbit ad ( ) X,let be two o-empty -ope subset o let have a dese orbit, thus the orbit o will eter both ad Let ad be the least itegers such that ad 82

5 Assume ad set The obviously Let ad be two maps, are said to be topologically cojugate, i there exists a homeomorphism such that, [2] Mappig which are cojugate are completely equivalet i terms o their dyamics ow we ca prove the ollowig lemma: Lemma 36: let be two cojugate uctio, the i is -trasitive the is -trasitive also let be two -ope subset o, sice is cotiuous ad oe to oe,the is *- cotiuous (Theorem33), Thus ad are -ope i Sice is -trasitive the there exists such that,ie, Thus is -trasitive uctio Reereces 1VKaa, agar,23, "Topological trasitivity or discrete dyamical systems"applicable mathematics i the golde age,arosa,pp LSBlock ad WACoppel,1991, "Dyamical i oe Dimesio" Lecture otes i Mathematics, Volume 1513,spriger-Verlag, Berli 3SKolyada ad L Soha,1997, "Some aspects o topological trasitive A survey"grazer Math Ber,334,pp3-35 4AlOmari A ad oorai S Md,29," ew characterizatio o compact spaces" proceedig o the 5 th Asia Mathematical coerece, Malaysia, pp53-6 5Majhool,Fatima ad Hamza, Sattar,211, "O -proper Actio" MSc thesis, Al Qadisiyah Uiversity 6Modak, Shyamapada,211 "Remarks o dese set" Iteratioal Math Forum,Vol6, o44, pp Degirmeci ad S Kocak,23,"Existece o a dese orbit ad topological trasitivity whe are they equivalet" Acta Math HugarVol99,o3,pp