Mixing, Weakly Mixing and Transitive Sets

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1 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 p-issn: wwwirjetet Mixig, Wealy Mixig ad Trasitive Sets Dr Mohammed Nohas Murad Kai Mathematics Departmet, Faculty o Sciece ad Sciece Educatio Uiversity o Sulaimai, Iraq *** Abstract - Topological δ-type trasitive uctio has Key Words: Topological δ-type trasitive uctio, called topological δ-type trasitive set ad ivestigate some o its topological dyamical properties Let A be a subset o a topological space (X, τ) The closure ad the iterior o A are deoted by Cl(A) ad It(A), respectively A subset A o a topological space (X, τ ) is said to be regular ope [] (resp preope [3]) i A = It(Cl(A)) (resp A It(Cl(A))) A set A X is said to be δ-ope [4] i it is the uio o regular ope sets o a space X The complemet o a regular ope (resp δ-ope) set is called regular closed (resp δ-closed) The itersectio o all δ-closed sets o (X, τ ) cotaiig A is called the δ-closure [4] o A ad is deoted by Cl (A) Recall that a subset S i a space X is topological dyamical properties called regular closed i S Cl ( It ( S )) A poit x ε X is bee itroduced by Mohammed Nohas Murad [] The aim o this paper is to ivestigate some properties o δtype trasitive sets i a give topological space ( X, ) I the preset paper, we studied some ew class o topological trasitive set called topological δ-type trasitive set ad ivestigate some o its topological dyamical properties called a δ-cluster poit [4] o S i S U or each CHAPTER FOUR 4 TOPOLOGICAL δ-type TRANSITIVE FUNCTION 4 INTRODUCTION Topological δ-type trasitive uctio has bee itroduced by Dr Mohammed Nohas Murad [] The aim o this paper is to ivestigate some properties o δ-type trasitive sets i a give topological space ( X, ) Further, we study ad ivestigate some properties o δ-type miimal mappig Moreover, the relatioships amog δ-type trasitive cojugated uctios ad the related classes o trasitive uctios are ivestigated The collectio o all δ-ope sets i a topological space ( X, ) orms a topology τs o X, called the semi-regularizatio topology o τs, weaer tha τ ad the class o all regular ope sets i τ orms a ope basis or τs Similarly, the collectio o all θope sets i a topological space ( X, ) orms a topology τθ o X, weaer tha τ Some types o sets play a importat role i the study o various properties i topological spaces May authors itroduced ad studied various geeralized properties ad coditios cotaiig some orms o sets i topological spaces, I the preset paper, we studied some ew class o topological trasitive set 05, IRJET regular ope set U cotaiig x The set o all δ-cluster poits o S is called the δ-closure o S ad is deoted by Cl (S ) A subset S is called δ-closed i δcl(s) = S The complemet o a δ-closed set is called δ-ope The amily o all δ-ope sets o a space X is deoted by O ( X, ) The δ-iterior o S is deoted by It (S) ad it is deied as ollows It (S ) {x X : x U It (Cl (U )) S,,U } A subset S o a topological space (X, τ ) is called δ-β-ope [5] i S Cl ( It (Cl (S ))) The complemet o a δ-β-ope set is called δ-β-closed [5] The itersectio o all δ-βclosed sets cotaiig S is called the δ-β-closure o S ad is deoted by Cl (S ) The δ-β-iterior o S is deied by the uio o all δ-β-ope sets cotaied i S ad is deoted by β Itδ(S) The set o all δ-β-ope sets o (X, τ ) is deoted by δβo(x) The set o all δ-β-ope sets o (X, τ ) cotaiig a poit x X is deoted by δβo(x, x) Fuctios ad o course irresolute uctios stad amog the most importat otios i the whole o mathematical sciece Various iterestig problems arise whe oe cosiders opeess Its importace is sigiicat i ISO 900:008 Certiied Joural Page 596

2 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 p-issn: wwwirjetet various areas o mathematics ad related scieces I 97, Crossley ad Hildebrad [6] itroduced the otio o irresoluteess May dieret orms o irresolute uctios have bee itroduced over the years I the preset paper, we deie ad itroduce some ew class o topological trasitive uctios called topological δ-type trasitive ad study some o its properties let as deote the collectio o all δ sets o a space (X; τ ), recall that τ= i ad oly i every ope set is closed ad thereore δ-type trasitive ad trasitive uctios are coicide We observe that or ay topological space (X, τ ) the relatio always holds We also have A Cl ( A) Cl ( A) Cl ( A) or ay subset A o X 4 PRELIMINARIES AND DEFINITIONS Deiitio4 Let A be a subset o a space X A poit x A is said to be a δ- Limit poit o A i or each δ ope set U cotaiig x, U ( A {x}) The set o all δ- limit poits o A is called the δ -derived set o A ad is deoted by Dδ (A) Deiitio4 [7] Let X be a topological space, a subset S o X is said to be regular ope (respectively regular closed) i It(clS) = S (respectively Cl(itS) = S) A poit x S is said to be a δ-cluster poit o S i U S, or every regular ope set U cotaiig x The set o all δcluster poit o S is called the δ-closure o S ad is deoted by Clδ (S) I Clδ (S) = S, the S is said to be δ-closed The complemet o a δ-closed set is called a δ-ope set For every topological space (X, τ), the collectio o all δope sets orm a topology or X which is weaer tha τ This topology has a base cosistig o all regular ope sets i (X, τ) Deiitio43 A uctio : ( X, ) (Y, ) is said to be β-irresolute i (V ) is β-ope i X or each β- ope set V o Y (see [3]) Deiitio44 Let (X, τ) be a topological space A subset A is called a locally closed set (briely LC-set) [], i A = U F, where U is ope ad F is closed Lemma45 [4] Let D be a subset o X The: () D is a δ-ope set i ad oly i It (D) D () Cl ( D c ) ( It ( D)) c ad It ( D c ) (Cl ( D)) c () Cl ( D) Cl ( D) or ay subset D o X ; () Every closed subset o X is δ-closed ad hece or ay subset D, Cl (D) is δ-closed Deiitio47 A uctio : ( X, ) (Y, ) is said to be every A, δ-cotiuous i cotiuous) i ad oly i or every δ-closed set A o (Y, ), ( A) C ( X, ) Deiitio48 A topological space ( X, ) is said to be δt [8] i or each pair o distict poits x, y o X there exists a δ-ope set A cotaiig x but ot y ad a δ-ope set B cotaiig y but ot x, or equivaletly, (X, τ ) is a δ-tspace i ad oly i every sigleto is δ-closed ([8], Theorem 5) Example49 Let (X, τ ) be a topological space such that X = {a; b; c; d} ad τ ={ϕ, X, {c}, {c, d},{a, b}, {a, b, c} Clearly, δo(x, τ )={ϕ, X, {a, b}, {c, d} Theorem49 I ad g are completely δ-β-irresolute, the g is completely δ-β- irresolute Theorem40 A uctio : ( X, ) (Y, ) is said to be δ-β-irresolute [5] (resp δ-β-cotiuous [9]) i (V ) is δ-β-ope (resp δ-β-ope) i X or every δ-βope (resp ope) subset V o Y Theorem4 For subsets A, B o X, the ollowig statemets hold: () D( A) D ( A) where D( A) is the derived set o A () I A B, the D ( A) D ( B) (3) D ( A) D ( B) D ( A B) ad D ( A B) D ( A) D ( B) Note that the amily o θ ope sets i ( X, ) always orms a topology o X deoted θ-topology ad that θtopology coarser tha τ We observe that or ay topological space ( X, ) the relatio always holds Note that θ-closed δ-closed closed α-closed 43 δ-types TRANSITIVE FUNCTIONS AND δ-minimal SYSTEMS By a topological system ( X, ) we mea a topological (4) or a ope (resp closed ) subset D o X, Cl ( D) Cl ( D) (resp It ( D) It ( D) ( A) A space X with A X a cotiuous uctio is called -ivariat i Lemma46 [4] I X is a regular space, the: 05, IRJET or ( A) O( X, ) or, equivaletly, is δ- together : X X A set (3) Cl ( D) Cl ( D) (resp It ( D) It ( D) or ay subset D o X [7] ISO 900:008 Certiied Joural Page 597

3 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: A dese orbit o a topological system o a set X is a orbit whose poits orm a dese subset o X Topologically trasitive ad existece o a dese orbit are two otios that play a importat role i every deiitio o chaos Propositio43 () Let ( X, ) be a topological space, ad : X X a cotiuous uctio, the is a topologically trasitive uctio i ad oly i or every pair o ope sets U ad V i X there is a positive iteger such that ( U) V () A uctio : X X is topologically trasitive i ( x) X or some x X Proo: Suppose that ( x) X or some x X The or every pair o o-empty, ope U, V X there are positive itegers m m m such that ( x) U ad ( x) V Hece ( U) V ad is topologically trasitive Recall that a topological system ( X, ) is said to be miimal i X has o closed -ivariat subset As well ow, I X is a metric space the the topological system ( X, ) is topologically trasitive i ad oly i it is miimal I X is a metric space, it is easy to see that i a topological system has a dese orbit, the it is trasitive, sice this orbit comes arbitrarily close to all poits The coverse is more diicult to prove For related wors see [] Recall that a system ( X, ) is said to be miimal i X does ot cotai ay o-empty, proper, closed -ivariat subset I such a case we also say that the uctio itsel is miimal I X is a metric space, it is easy to see that i a topological system has a dese orbit, the it is trasitive, sice this orbit comes arbitrarily close to all poits The coverse is more diicult to prove For related wors see [] Recall that a system ( X, ) is said to be miimal i X does ot cotai ay o-empty, proper, closed - ivariat subset I such a case we also say that the uctio itsel is miimal Give a poit x i a system, ( X, ), O ( ) {, ( ), x x x ( x),}, deotes its orward orbit (by a orbit we mea a orward orbit, ad (x) deotes its -limit set, ie the set o limit poits o the sequece x, ( x), ( x), Deiitio43 [3] (δ-type waderig poits): Let ( X, ) be a topological system A poit x X is δ-type waderig or a uctio i it belogs to a δ-ope set U disjoit rom ( U) or all 0 The set o all δ-type waderig poits is δ-ope ivariat set : its complemet, the set o all o-waderig poits is a δ-closed ivariat set Let me itroduce a ew deiitio o miimal uctios called δ-miimal ad we study some ew theorems associated with this ew deiitio A system ( X, ) is called δ-miimal i X does ot cotai ay o-empty, proper, δ-closed -ivariat subset I such a case we also say that the uctio itsel is δ- miimal Aother deiitio o miimal uctio is that i the orbit o every poit x i X is dese i X the the uctio is said to be miima Let us itroduce ad study a equivalet ew deiitio Deiitio433[3] (δmiimal) Let X be a topological space ad be δ-irresolute uctio o X The ( X, ) is called δ-miimal system (or is called δ-miimal uctio o X) i oe o the three equivalet coditios hold: () The orbit o each poit i X is δ-dese i X () Cl ( O ( x)) X or each x є X (3) Give x є X ad a oempty δ-ope U i X, there exists є N such tha ( x) U Theorem434[3] For ( X, ) the ollowig statemets are equivalet: () is a δ-miimal uctio () I E is a δ-closed subset o X with ( E) E, we say E is ivariat The E= or E=X (3) I U is a oempty δ-ope subset o X, the 0 Proo: ( U ) X () (): I A, let x є A Sice A is ivariat ad δ- closed, ie Cl ( A) A so Cl ( O ( x)) A O other had Cl ( O ( x)) X Thereore A = X () (3) Let A=X\ ( U ) Sice U is oempty, A 0 X ad Sice U is δ-ope ad is δ-cotiuous, A is δ- closed Also ( A) A, so A must be ϕ is δ-cotiuous, A is δ-closed Also must be a empty set ( A) A, so A 05, IRJET ISO 900:008 Certiied Joural Page 598

4 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 p-issn: wwwirjetet (3) (): Let x є X ad U be a oempty δ-ope subset o X Sice x є X = some 0 So 0 (U ) Thereore x є (U ) or ( x) U Propositio 435[3] Let X be a δ-compact space without isolated poit, i there exists a δ-dese orbit, that is there exists x0 X such that the set O ( x0 ) is δ- dese the is δr-cojugate i there is a δr-homeomorphism h : X Y such that h g h We will call h x 0 be such that OF ( x0 ) is δ-dese Give ay topological δ- Cojugacy Thus, the two topological systems with their respective uctio actig o them share the same dyamics Now, we proceed to prove a importat propositio: Propositio438 (A) i : X X ad g : Y Y are topologically δ-trasitive Proo Let ψ(y) is dese i X Thus, i oe ids a topological Cojugacy o a uctio with a simpler uctio g, oe ca aalyze the simpler uctio g to obtai iormatio about dyamical properties o the origial uctio For related wors see [0] Deiitio 437 Two topological systems : X X ad g : Y Y are said to be topologically pair U, V o δ-ope sets, by δ-desity there exists such that ( x0 ) U, but OF ( x0 ) is δ- dese implies that OF ( ( x0 )) is δ-dese, there exists m such that m ( ( x0 )) V Thereore m ( x0 ) m (U ) V That is m (U ) V So is topological δ-trasitive Deiitio436 A system ( X, ) is called topologically δmixig i or ay o-empty δ-ope set U, there exists N such that (U ) is dese i X N As well ow i every poit x i the space X has a dese orbit, ad the X is mixig I two topological spaces are homeomorphic, the ot oly their respective sets o poits, but also their collectios o ope sets are i a oe-to-oe correspodece Homeomorphisms show us whe two topological spaces should be cosidered to be the same i the eye o topology Topologically δ-cojugatig: Oe usual way to relate two topological systems : X X ad g : Y Y is with topologically δr-cojugated by δr-homeomorphism h : X Y The () T is δ- trasitive set i X i ad oly i h(t) is δtrasitive set i Y; () T is δ mixig set i X i ad oly i h(t) is δ mixig set i Y (B) I h is ot δr -homeomorphism but oly δ -irresolute surjectio (a semi- δ r -Cojugacy), the i Tis topological δtrasitive set i X the h(t) is topological δ- trasitive set i Y Deiitio439 Let :X X uctio o a topological space X A udametal δ-type domai or orbit o is δ-ope subset Cl (D) i at least oe poit Propositio430 Let h g h, where ad g are called the dyamic udametal the homeomorphism ψ : Y X, the or all y Y the orbit Og ( y) is dese i Y i ad oly i the orbit O ( ( y)) o 05, IRJET D X such that every itersect D i at most oe poit ad itersect the topological otio o Cojugacy o Two topological systems (X, ) ad (Y, g) are topologically δ-cojugated i there exist a δr-homeomorphism h : X Y such that uctios that act o the spaces X ad Y respectively, Cojugacy show us whe two topological systems should be cosidered to be the same i the eye o topological dyamics the orbit o two topological systems (X, ) ad (Y, g) are topological cojugate or cojugate i there exists a homeomorphism h X Y such that h((x))=g(h(x)) or each x i X Topological cojugacy requires that orbits be put ito oe-to-oe correspodece ad preserves may topological dyamical properties For example, I : X X ad g : Y Y are topologically cojugated by be δ-irresolute sel- : X X ad g : Y Y be two δ-irresolute sel-uctios Assume that there are a δ-type domai D X or, a udametal δ-type domai Dg Y or g ad a δr- homeomorphism such that ad g are h : Cl ( D ) Cl ( Dg ) g h h o (Cl ( D )) Cl ( D ) The topologically δr-cojugated For the sae o compariso, let us itroduce ad deie topological otios o δ-type recurrece A poit x X is δtype o-waderig i or ay δ-ope U X cotaiig x, there exists N 0 such that N (U ) U The uctio is δ-type o-waderig i every poit o X is δ-type o-waderig A uctio ISO 900:008 Certiied Joural is δ-type trasitive i or Page 599

5 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: ay oempty δ- ope sets U, V X, there is a N 0 such that N ( U) V A uctio is topologically δ- type mixig i or ay oempty δ- ope sets U, V X, there is a N 0 such that ( U) V or all N Deiitio43 Let ( X, ) be a topological system A poit x ϵ X is δ-type o-waderig i or ay δ-ope set U cotaiig x there is N > 0 such that N ( U) U The set o all δ-type o-waderig poit is deoted by NW δ( ) A poit which is ot δ-type o-waderig is called δ-type waderig We ca easily prove the ollowig Propositio: Propositio43 Let ( X, ) be a topological system o δ-hausdor space X The: ( i) NW ( ) is closed ( ii) NW ( ) is ivariat (iii) I is ivertible, the NW ( ) NW ( ) (iv) I X is compact the NW ( ) (v) I x is type o-waderig poit i X, the or every ope set U cotaiig x ad 0 ϵ N there is such that ( U) U 0 Remar 433 Ay δ-dese subset i X itersects ay δ- ope set i X Proo: Let A be a δ-dese subset i X, the by deiitio, Cl ( A) X, ad let U be a oempty δ-ope set i X Suppose that A U=ϕ Thereore c A U =B So (A) c B U is δ-closed ad Cl Cl (B), ie Cl (A) B, but Cl ( A) X, so X B, this cotradicts that U Associated with the ew deiitio o topologically δ-type trasitive we ca prove the ollowig ew theorem Theorem434 Let ( X, ) be a regular space ad : X X be δ-irresolute uctio The the ollowig statemets are equivalet: () is δ-type trasitive uctio () For every oempty δ-ope set U i X, ( U ) is δ- dese i X 0 (3) For every oempty δ-ope set U i X, ( U ) is δ- dese i X 0 (4) I B X is δ-closed ad B is orward - ivariat ie ( B) B the B=X or B is owhere δ-dese (5) I U is δ-ope ad ( U) U the U= or U is δ- dese i X We have to prove this theorem: Proo: () () Assume that ( U ) is ot δ-dese The there exists a 0 o empty δ-ope set V such that implies that 0 ( U ) V This ( U) V or all є N This is a cotradictio to the δ-type-trasitivity o Hece ( U ) is δ-dese i X 0 () () Let U ad V be two oempty δ-ope sets i X, ad let ( U ) be δ-dese i X, this implies that ( U ) V 0 0 by Remar 433 This implies that there exists m є N such that m U) V trasitive uctio () (3) ( Hece is topologically δ-tye It is obvious that ( U ) is δ-ope ad sice is δ- 0 type trasitive uctio, it has to meet every δ-ope set i X, ad hece it is δ-dese (3) () Let E ad F be two δ-ope subsets i X The ( F) is δ-dese, this implies that 433This implies m F 0 0 ( F) E, by Remar that there exists m є N such that m m ( F) E Thereore, ( ( F) E) m (E) So is δ-type trasitive () (4) Suppose is δ-type trasitive uctio, E X is δ- closed ad ( E) E Assume that E ad E has a oempty δ-iterior (ie it ( E ) ) I we deie V=X\E so V is δ-ope because V is the complimet o δ- closed Let it ( E ) We have F E be δ-ope sice ( F) E sice E is ivariat, thereore ( F) V, or all є N This is a 05, IRJET ISO 900:008 Certiied Joural Page 600

6 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 p-issn: wwwirjetet cotradictio to topological δ-type trasitive Hece E=X or E is owhere δ-dese (4) () Let V be a oempty δ-ope set i X Suppose is ot a topological δ-type trasitive uctio, rom (3) o this theorem (V ) is ot δ-dese, but δ-ope Deie 0 E X \ (V ) which is δ-closed, because it is the 0 complemet o δ-ope, ad E X Clearly ( E ) E Sice (V ) 0 exists is ot δ-dese so by Remar 433, there o-empty δ-ope W i X such that (V ) W This implies that W E This is a 0 a cotradictio to the act that E is owhere δ-dese Hece is a topological δ-type trasitive uctio itersects U This holds or ay o-empty δ- ope set U ad thus shows that O (h( y)) is δ-dese The other implicatio ollows by exchagig the role o ad g Theorem436 Ay two δ-miimal sets must have empty itersectio Proo: Let M ad M be two distict δ-miimal sets, ad suppose that A M M The A is δ-closed, ad or every a A ad every N, (a) M M, so A is ivariat But the A is a proper subset o both M ad M which is δ-closed, ivariat ad o-empty, cotradictig the act that M ad M are δ-miimal Lemma437 I ( X, ) ad (Y, g ) are tow topological systems The set o periodic poits o g is δ-dese i X Y i ad oly i, or both o ad g, the sets o () (5) periodic poits i X ad Y are δ-dese i X, respectively Y Lemma 438 Let : X X ad g : Y Y be uctios Suppose that the uctio is δ-type trasitive, U X is ad assume that the product g is topological δ-type δ-ope ad (U ) U Assume that U ad U is ot trasitive o X Y The the uctios ad δ-dese i X (ie Cl (U ) X ) The there exists a o- empty δ-ope V X \ Cl (U ) sice Cl (U ) is δ-closed, such that U V Further (U ) V or all є N This implies U (V ) or all є N, a cotradictio to beig δ-type trasitive uctio Thereore U or U is δ-dese i X Propositio 435 i : X X ad g : Y Y are topologically δr-cojugated by the δr-homeomorphism h :Y X The or all y Y the orbit i Y i ad oly i the orbit Og ( y) is δ-dese O (h( y)) o h(y) is δ-dese i X Proo: Let h :Y X be the δr- Cojugacy Assume that is δ-dese ad let us show that Og ( y ) O (h( y)) is δ-dese For ay U X o-empty δ-ope set, h (U ) is a δ-ope set i Y sice h is δ-irresolute because h is a δrhomeomorphism ad it is o-empty sice h is surjective By δ-desity o Og ( y), there exists N such that g ( y) h (U ) h ( g ( y)) U Sice h is a δrcojugatio the so (h( y)) h( g ( y)) U, 05, IRJET h h g thereore O (h( y)) g are both topological δ-type trasitive o X ad Y respectively Deiitio439 Let : X X be a uctio o the topological space X ad A is a closed ivariat subset o X I or every oempty δ-ope sets U, V X, such that A U ad A V, there exists a positive iteger 0 such that or every 0, (U, ) V the A is called topologically δ-type mixig set It is clear that topological δ-type mixig implies topological δ-type trasitive set but ot coversely There is a eve stroger otio that implies topological δ-type mixig Deiitio430 Let : X X be a uctio o the topological space X I or every oempty δ -ope subset U X there exist a positive iteger 0 such that or every 0, (U ) X, the is called locally δ-type evetually oto Note that locally δ-type evetually oto δ-- type mixig wealy δ--mixig δ--type trasitivity Lemma43 The product o two topologically δ-type mixig uctios is topologically δ-type mixig Proo: Let ( X, ), (Y, g ) be topological systems ad, g be topologically ISO 900:008 Certiied Joural δ-type mixig uctios Give Page 60

7 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: W W X, there exists δ-ope sets, Y U, U X ad V, V Y, such that U V W adu V W By assumptio there exist ad such that ( U) U or ad g ( V ) V or For 0 max{, } We get [( x g ) ( U x V )] ( U x V ) [ ( U ) x g ( V )] ( U x V ) [ ( U) U ] x [ g ( V ) V ] where ( x, ) Cl { ( x) : } Which meas that g is topologically δ-type mixig 0 The term wea icompressibility seems to have appeared Deiitio435 A set A X is wealy δ-type irst i [] ad we adopt this term i this sectio: icompressible (or has wea δ-type icompressibility) Deiitio43[3] (Wealy Icompressible) a set i D Cl( ( A\ D)) wheever D is a oempty, δ- A X is wealy icompressible i or ay proper oempty subset U A which is ope i A, closed, proper subset o A Clearly A is wealy δ-type icompressible i ad oly i Cl ( ( U)) ( A\ U) or Cl ( ( U)) ( A \ U) Equivaletly we ca say that or ay proper, oempty subset U A which is δ-ope i A ay o-empty closed subset D A we have that Lemma436 Every wealy icompressible is wealy δ- D Cl( ( A\ D)) type icompressible but ot coversely Lemma433 [3] I A x ) or : X X ad ( 0 x 0 X the A is wealy icompressible Proo: Assume that or some closed M A we have that M Cl( ( A\ M)), the by ormality there are ope sets U ad V such that Cl( U) Cl( V), M U ad, Cl( ( A\ M)) V Thus ( A \ M ) ( V ) W, where W is ope by cotiuity ( Cl( W)) Cl( ( W)) Cl( V), so ( Cl( W)) Cl( U) Sice A ( x 0 ) ( W U), there is a iteger 0 0 such that ( x 0 )W U or every 0 Moreover, ( x 0 ) W or iiitely may ad m ( x 0 ) U or iiitely may m 0, so or iiitely may 0, ( x 0 ) W ad ( x 0 ) U Thus there is a sequece { } such that i ( x 0 )W, i ( x 0 ) U ad i i N i y lim ( x0) Cl( W) i The i i y) (lim ( x )) lim ( x ) Cl( ); ( 0 0 U i i ie ( y) ( Cl( W)) Cl( U), which cotradicts the act that ( Cl( W)) Cl( U) Hece M Cl( ( A\ M)) We deie ew deiitios o stable subsets o topological spaces Deiitio434 Let set 0 A X be δ- compact ivariat - A is δ-type stable i every δ-ope set U cotaiig A cotais δ-ope set V cotaiig A such that ( V ) U or all 0 - The δ -closed, oempty ivariat set A is said to be δ - type attractor i ad oly i there is δ -ope set U A such that (i) Cl ( ( U)) U, (ii) ( x, ) A or every x U 44 NEW TYPES OF TRANSITIVE FONCTIONS ON SEMI-REGULAR SPACES Deiitio44 Let X be a set, ad B a collectio o subsets o X Recall that B is a basis or a topology o X, ad each B i B is called a basis elemet i B satisies the ollowig axioms: B For every x X, there is a set B B such that x B, ad x B B B I B ad B are i B ad, the there exists a B 3 i B such that x B3 B B Deiitio44 Let X be a topological space ad let x X A eighborhood o x is a ope set N such that x N Propositio443 Let X is a set ad B a basis over X Let be a collectio o subsets o X which icludes ф ad which is closed with respect to arbitrary uios o basis elemets The is a topology over X Deiitio444 Let X be a set ad B a basis over X The topology geerated by B is created by deiig the ope sets o to be ф ad all sets that are a uio o basis elemets Give a space ( X, ), deote by the topology o X whose basis cosists o regular ope subsets o ( X, ) The space ( X, ) is said to be the semi-geeralizatio o ( X, ) It is easy to see that ad that the 05, IRJET ISO 900:008 Certiied Joural Page 60

8 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: spaces ( X, ) ad ( X, ) have the same regular ope subsets Deiitio445 A space whose regular ope subsets orm a base or its topology is called semi-regular The set o all ope subsets o semi-regular space are deoted by O sr (X ), whose elemets are called sr-ope sets Deiitio446 Let ( X, ) ad ( Y, ) be two topological spaces, A uctio is called sr-irresolute i or every sr-ope subset H o Y, ( H) is sr-ope i X Deiitio447 Let ( X, ) be a topological space, : X X be sr- irresolute uctio, the the map is called semi-regular trasitive ( i short: sr-trasitive) i or every pair o o-empty sr-ope sets U ad V i X there is a positive iteger such that ( U) V I this case, we ca say that the system (X, ) is sr-trasitive It is srmiimal i every orbit is sr-dese Deiitio 448 () A poit x X is sr-recurret i, or every sr-ope set U cotaiig x, iiitely may N satisy ( x) U () Let (X, τ) be a topological space, : X X be srirresolute map, the the uctio is called topologically sr- strogly mixig i, give ay oempty sr-ope subsets U, V X N such that ( U) V or all N A subset B o X is -ivariat i B) B ( A oempty sr-closed ivariat subset B o X is sr- miimal, i Cl ( O ( x)) B or every x B A poit x X is sr-miimal sr i it is cotaied i some sr-miimal subset o X Clearly i is topologically sr- strogly mixig the it is also sr-trasitive but ot coversely (3) The uctio is sr-exact i, or every oempty srope set U X, there exists some N such that ( U) X Note that topological sr-exactess implies 45 S-REGULAR-CHAOS AND TOPOLOGICAL S- sr-mixig implies wealy sr-mixig implies sr-trasitivity (4) The uctio is (topological) semi-regular trasitive (resp, sr-mixig) i or ay two oempty sr- ope sets U, V X, there exists some N such that ( U) V (resp m ( U) V, or all m ) (5) The uctio is wea sr-mixig i is sr type trasitive o X X (6) The sr-mixig uctio : X X is pure sr-mixig i ad oly i there exists sr- ope set U X such that ( U) X or all 0 (6) The uctio : X X is sr-type chaotic i is srtrasitive o X ad the set o periodic poits o is srdese i X (7) The uctio is called sr-type exact chaos (resp, srtype mixig chaos ad wealy sr-type mixig chaos) i is sr-exact (resp, sr-mixig ad wealy sr-mixig) ad srtype chaotic uctio o the space X (8) Let ( X, ) be a topological space, : X X be srirresolute uctio, the the set A X is called semiregular trasitive set ( i short sr-trasitive set) i or every pair o o-empty sr-ope sets U ad V i X with A U ad AV there is a positive iteger such that ( U) V (9) Let ( X, ) be a topological space, : X X be srirresolute uctio, the the set A X is called topologically sr-mixig set i, give ay oempty sr-ope subsets U, V X with A U ad AV the N 0 such that ( U) V or all N A is called a wealy sr- ( X, i or ay choice o oempty sr -ope (0) The sr-closed set X mixig set o ) subsets V,V o A ad oempty sr ope subsets U, U o X with A U ad A U there exists N such that ( V ) U ad ( V ) U Note: I A is a wealy sr- mixig set o ( X, ), the A is a semi-regular trasitive set o ( X, ) Deiitio 449 Two topological dyamical systems (X, ) ad (Y, g) are said to be semi regular -cojugated (i short sr-cojugate) i there is a semi regular homeomorphism (i short srhomeomorphism) h : X Y such that h g h REGULAR-CONJUGACY I itroduce ad deie sr-type trasitive uctios ad sr type miimal uctios I will study some o their properties ad prove some results associated with these ew deiitios I ivestigate some properties ad characterizatios o such uctios Let (X, ) be a topological system A uctio : X X is called semi-regular chaotic, i it is topological sr-type trasitive ad, its periodic poits are sr-dese i X, ie every oempty sr-ope subset o X cotais a periodic poit (A poit x X is called periodic i there exists with ( x) x The set o all periodic poits o deoted by Per ( ) 05, IRJET ISO 900:008 Certiied Joural Page 603

9 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: Deiitio 45 Recall that a subset A o a space X is called sr-dese i X i Cl sr (A), we ca deie equivalet deiitio that a subset A is said to be sr-dese i or ay x i X either x i A or it is a sr-limit poit or A Remar 45 ay sr-dese subset i X itersects ay srope set i X Deiitio 453 Recall that a subset A o a topological space (X, τ) is said to be owhere sr-dese, i its sr-closure has a empty sr-iterior, that is, It ( Cl ( A)) sr sr ( Deiitio 454 i or x X the set { x) : N} is srdese i X the x is said to have sr-dese orbit I there exists such a x X, the is said to have sr-dese orbit Deiitio 455 A uctio : X X is called s- regular-homeomorphism i is sr-irresolute bijective ad : X X is sr-irresolute Deiitio 456 Two topological systems : X X, x ( x ) ad g : Y Y,, y g( y ) are said to be topologically s-regular -cojugate i there is s- regular -homeomorphism h : X Y such that h g h We will call h a topological semi-regular Cojugacy (i short sr-cojugacy) Whe the two systems are topologically s-regular -cojugate they have the same dyamics The I have proved some o the ollowig statemets: h : Y X is a topological s-regular-cojugacy h g h N 3 x X is a periodic poit o i ad oly i h(x) is a periodic poit o g 4 I x is a periodic poit o the uctio with stable set W (x), the the stable set o h(x) is ( W ( x)) h 5 The periodic poits o are dese i X i ad oly i the periodic poits o g are dese i Y 6 The uctio is sr-exact i ad oly i g is sr-exact 7 The uctio is sr-mixig i ad oly i g is sr-mixig 8 The uctio is semi- regular chaotic i ad oly i g is semi-regular chaotic 9 The uctio is wealy sr-mixig i ad oly i g is wealy sr- mixig Remar 457 I {x 0, x, x, } deotes a orbit o x ( x ) the y = h(x ), y y = h(x ), } = h(x ) = h((x { 0 0 y = h(x ), yields a orbit o g sice )) = g(h(x ) = g(y + + ie ad g have the same id o dyamics I itroduced ad deied the ew type o trasitive called topologically semi-regular trasitive (i short sr-trasitive ) ), i such a way that it is preserved uder semi-regular cojugatio Theorem 458 For sr -irresolute uctio : X X, where X is a topological space, the ollowig are equivalet: is semi-regular trasitive; Ay Proper sr closed subset A X ( A) A is owhere sr dese; 3 A X ( A) A, A is either sr dese or owhere sr dese; 4 Ay subseta X ( A) A with oempty sr iterior is sr dese Propositio 459 i : X X ad g : Y Y are sr -cojugated by the sr homeomorphism h : Y X The or all y Y the orbit O g (y) is sr-dese i Y i ad oly i the orbit O ( h( y)) o h(y) is sr -dese i X Propositio 450 Let X be a sr-compact space without isolated poit, i there exists a sr-dese orbit, that is there exists x 0 X such that the set O ( x 0 ) is sr- dese the the uctio is semi-regular trasitive Propositio 45 i : X X ad g : Y Y are sr -cojugated by h : X Y The: () is semi-regular trasitive i ad oly i g is semiregular trasitive () is sr -miimal i ad oly i g is sr -miimal (3) is topologically sr -mixig i ad oly i g is topologically sr -mixig Propositio 45 I the two uctios : X X ad g : Y Y are topologically semi-regolar-cojugated by h rom X to Y The () The uctio is sr -exact i ad oly i g is sr -exact () is wealy sr -mixig i ad oly i g is wealy sr - mixig (3) is sr -type chaotic o X i ad oly i g is sr -type chaotic i Y Proo () ) give ay oempty sr -ope set V i Y, let us tae U h ( V ) Clearly, U is a oempty sr -ope set i X Sice is sr -exact, there exists N such that ( U) X Notig that ad g are sr -cojugate uctios ad that h is a sr-homeomorphism, it ollows that g ( V ) g ( h( U)) h( ( U)) h( X ) Y Sice V is a arbitrary sr -ope set, this implies that g is sr -exact ) It ca be proved similarly Proo() 05, IRJET ISO 900:008 Certiied Joural Page 604

10 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: ) For ay two oempty sr-ope sets V, W Y Y, accordig to the costructio o product topology, it ollows that there exist oempty sr-ope sets W, W, V V Y such that W W W V, V V ad Sice is wealy sr-mixig, t9\here exists N such that ( ) ( h ( W ) h ( W )) ( h ( V ) h ( V )), This implies that ( g g) ( W ) V ( g g) ( W W ) ( V V ) ( g g) [( h h)( h ( h h)( ( h h)([ ( W ) h ( h ( W )) ( h ( h ( W )) ( h ( W ))] [( h h)( h ( W ))) [( h h)( h Thereore, g is wealy sr-mixig ) It ca be proved similarly ( W ))] ( h ( V ) h ( V ) h ( V ) h ( V ))] ( V ))] ( V ))) Proo (3) Necessity: Similarly to the proo o Propositio 45 part (), it ca be veriied that g is semi-regular trasitive o Y, as is semi-regular trasitive o X Accordig to the deiitio o periodic poits, it is easy to chec that Per( g) h( Per( )) Applyig this, oe has Cl ( Per( g)) Cl [ h( Per( ))] h[ Cl ( Per( )] h( X) Y sr sr This implies that Cl sr [ Per( g)] Y Thereore g is sr-type chaotic uctio o Y ad also Cl sr sr [ Per( g)] Y, it ollows that g is sr-type chaotic Suiciecy ca be proved similarlys5 46 SEMI REGULAR-CHAOS IN PRODUCT TOPOLOGICAL SPACES Give two topological systems ( X, ),( Y, g) Cosider the product o the uctios ad g as g : X Y X Y, deied by ( g)(x, y) ( (x), g(y)), with product topology o X Y Lemma 46 Let ( X, ),( Y, g) be topological systems The set o periodic poits o g is sr-dese i X Y i ad oly i, or both o ad g, the sets o periodic poits i X ad Y are sr-dese i X, respectively Y Proo: Assume that the set o periodic poits o is srdese i X (ie Cl sr ( Per( )) X ) ad the set o periodic poits o g is sr-dese i Y (ie Cl sr ( Per( g)) Y ) We have to prove that the set o periodic poits o g is sr-dese i X Y Let W X Y be ay o-empty -ope set The there sr exist o-empty sr-ope sets U X ad V Y with U V W By assumptio, there exists a poit x U such that ( x) x with Similarly, there exists y V such that g m ( y) y with m For p (x, y) W ad m we get ( g) ( p) ( g) (( ( x, y) ( x), g ( y)) ( x, y) Thereore W cotais a periodic poit ad thus the set o periodic poits o g is sr-dese i X Y ) Coversely let U X ad V Y be o-empty sr - ope subsets The U V is a o-empty sr-ope subset o X Y As the set o the periodic poits o g is srdese i X Y, there exists a poit p (x, y) U V such that ( g) ( x, y) (( ( x), g ( y)) ( x, y) or some From the last equality we obtai x U ad g ( y) y or y V p ( x) x or some The semi-regulardeseess o periodic poits carries over rom actors to products But, topological semi-regular trasitivity may ot carry over to products The coverse o this situatio is however true: Lemma 46 Let : X X ad g : Y Y be uctios ad assume that the product g is semiregular trasitive o X Y The the uctios ad g are both topological semi-regular trasitive o X ad Y respectively Proo We have to prove the semi-regular trasitivity o ; the semi-regular trasitivity o g ca be proved similarly Let U be o-empty sr-ope sets i X,U U U Y ad V V Y are sr-ope i X Y As g is semi-regular trasitive, there exists The the sets a positive iteger such that From the equalities: ( g) ( U ) V [ ( U ) g ( x g) ( U) V ( Y )] [ V Y ] [ ( U) V ] [ g ( Y ) Y ], Thus is semi-regular trasitive uctio Deiitio 463 Let : X X be a uctio o the topological space X I or every oempty sr-ope subsets U, V X there exists a positive iteger 0 such that 05, IRJET ISO 900:008 Certiied Joural Page 605

11 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: or every 0, ( U,) V the is called () is wealy sr-mixig i ad oly i g is wealy srmixig topologically sr- mixig (3) is semi-regular chaotic o X i ad oly i g is semiregular chaotic i Y It is clear that topological sr-mixig implies semi-regular trasitive There is a eve stroger otio that implies Lemma 474 Let ( X, ), ( Y, g) be topological systems topological sr- mixig Deiitio 464 Let : X X be a uctio o the The set o periodic poits o g is sr-dese i X Y i space X I or every oempty sr-ope subset U X ad oly i, or both o ad g, the sets o periodic poits i X ad Y are sr-dese i X, respectively Y there exists 0 N \{0} such that or every, Lemma 475 Let : X X ad g : Y Y be 0, ( U) X, the is called semi-regular uctios ad assume that the product g is semiregular trasitive o X Y The the uctios ad g exact Lemma 465 The product o two topologically sr-mixig are both semi-regular trasitive o X ad Y respectively uctios is topologically sr-mixig Lemma 476 The product o two topologically sr-mixig Proo Let ( X, ), ( Y, g) be topological dyamical uctios is topologically sr- mixig systems ad, g be topologically sr- mixig uctios Theorem477 Let : X X ad g : Y Y be semiregular chaotic ad topologically sr-mixig uctios o Give W, W X Y, there exists sr-ope sets U, U X ad V, V Y, such that U V W ad topological spaces X ad Y The g : X Y X Y, is semi-regular chaotic U V W By assumptio there exist ad Propositio478 Every trasitive set is a δ-type such that trasitive set, but ot coversely, uless, the space is ( U ) U or ad g ( V ) V regular or For 0 max{, } Propositio479 Every topologically mixig set is a δ- we get mixig set but ot coversely, uless the space is regular [( x g) ( U x V )] ( U x V ) [ ( U) x g ( V )] ( U x VPropositio470 ) Every topologically alpha-trasitive set is trasitive set bot ot coversely [ ( U ) U ] x [ g ( V ) V ] Which meas that g is sr- mixig We give some suiciet coditios or a product uctio to be semi-regular chaotic Theorem 466 Let : X X ad g :Y Y be semi-regular chaotic ad topologically sr- mixig uctios o topological spaces X ad Y The g : X Y X Y, is semi-regular chaotic Proo: The uctio g has sr-dese periodic poits by Lemma 46 ad it is topologically sr- mixig by Lemma 465 ad hece semi-regular trasitive Thus the two coditios o semi-regular chaos are satisied 47 CONCLUSION : There are the ollowig results: Propositio 47 Semi - regular exact chaos semi-regular-mixig chaos wea semi-regular - mixig chaos semi-regular chaos Propositio 47 i is topologically sr-mixig the it is also semi-regular trasitive but ot coversely We ca easily prove the ollowig Propositio Propositio 473 I : X X ad g : Y Y are topologically semi regular -cojugate The () The uctio is semi-regular exact i ad oly i g is semi-regular exact 05, IRJET ISO 900:008 Certiied Joural Page 606 REFERENCES [] Mohammed Noha Murad Kai, New Types o δ- Trasitive Maps Iteratioal Joural o Egieerig & Techology IJET-IJENS, Vol: No06 (03) [] M Stoe, Applicatios o the theory o Boolea rigs to geeral topology, Tras Amer Math Soc, 4(937) [3] A S Mashhour, M E Abd El-Mose ad S N El-Deep, O Pre-cotiuous ad wea pre-cotiuous maps, Proc Math Phys Soc Egypt, 53(98), [4] N V Veli co, H-closed topological spaces Amer Math Soc, Trasl 78(968), 0-8 [5] S Jaari ad N Rajesh, O δ-β-irresolute uctios (uder preparatio) [6] S G Crossley ad S K Hildebrad, Semi topological properties, Fud Math 74(97)33-54 [7] T Noiri, O δ-cotiuous uctios, J Korea Math Soc6 (979/80) No,6-66 [8] M Caldas ad S Jaari, O δd-sets ad associated wea separatio axioms Bull Math Sci Soc () 5(00), No, 73{85 [9] E Hatir ad T Noiri, O δ-β-cotiuous uctios, to

12 Iteratioal Research Joural o Egieerig ad Techology (IRJET) e-issn: Volume: 0 Issue: 08 Nov-05 wwwirjetet p-issn: appear i Chaos, Solutios ad Fractals [0] L Bloc ad EM Cove, Topological Cojugacy ad trasitivity or a class o piecewise mootoe maps o the iterval, Tras Amer Math Soc 300 (987), [] Silverma S "O maps with dese orbits ad the deiitio o chaos," Rocy J Math,, (99), [] F Bali Brea ad C La Paz A characterizatio o the ω- limit sets o iterval maps Acta Math Hugar, 88(4):9 300, 000 Dr Mohammed Nohas Murad is a assistat proessor i Faculty o sciece ad sciece educatio, Uiversity o Sulaimai He holds MSc ad PhD i Mathematics ad DSc i Sciece He is iterested i Catastrophe theory, Topological dyamical systems, measure theory ad sustaiable developmet o mathematical sciece [3] Mohammed Nohas Murad Kai, Itroductio to topological dyamical systems, boo with ISBN: , SciecePG, New Yor, USA 05, IRJET ISO 900:008 Certiied Joural Page 607