It J Pure App Sci Techo 6( (0 pp7-79 Iteratioa Joura of Pure ad Appied Scieces ad Techoogy ISS 9-607 Avaiabe oie at wwwijopaasati Research Paper Reatioship Amog the Compact Subspaces of Rea Lie ad their Chaotic Properties Payer Ahmed * M Rahma ad S Kawamura Departmet of Mathematics Jagaath Uiversity Dhaa Bagadesh Departmet of Mathematics Facuty of Scieces Yamagata Uiversity Yamagata 990 Japa * Correspodig author e-mai: (drpayerahmed@yahoocom (Received: 4-8-; Accepted: -8- Abstract: We show a homeomorphic equivaet reatio amog the compact subspaces (Cator midde-third sets two-sided shift map of Cator sets ad terary sets etc of the rea ie Usig such resut we show that there eists a chaotic homeomorphism of a compact subspace of the rea ie oto itsef if ad oy if it is homeomorphic to the Cator set Keywords: Cator sets Shift map Terary sets HomeomorphismChaotic maps Itroductio: Though the cocept of Cator sets is a property of geera spaces we have discussed here oy i R the sets rea ie A subset X of R that cotais o iterva is said to be totay discoected This defiitio is equivaet to that for a i X the coected compoet of deoted by C( is equa to {} Here we ote that coected compoet is the argest coected set cotaiig itsef A poit is a isoated poit of a subset X of R if there eistsε > 0 such that U ( ε X cosists sige poit Cator Sets ad Terary Sets with their Reatios Defiitios: A compact subspace of X of R is said to be a Cator set if X id totay discoected ad have o isoated poits Let I [0] I [0/ ] I [/ ] I [0/ 9] I [/9/9] I [6/97/9] 0 0 I 4 [8/9] ad so o The sequece { I : 0 ad } is epressed as foows by the mathematica iductio:
It J Pure App Sci Techo 6( (0 7-79 7 (i I 0 [0] (ii Suppose I is epressed as I [ ] The I ad I are epressed as I [ ] respectivey We put C ( R ( I The set C is I U 0 caed the Cator midde-third set For [0] the epressio where is i either 0 or is the terary epasio of rea umber of For eampe the sequece 000 is the terary epasio of ¼ sice 0 0 4 4 L ( ( / ad 9 the sequece 0000 ad 00 are the terary epasios of / sice 0 0 L / ad 0 L / Aso the sequece 0000 is the terary epasio of / sice 0 0 L / We put { D R : {0}} The set D is caed the terary set Lemma The Cator midde third set C ad the terary set D are equa Proof: Let D We show that U I for a 0 We show it by the theory of mathematica iductio: assume that foows: U I (i For 0 Triviay we have I 0 [0] (ii We ie I for some the the umber is epressed as m m m m m m m m m We put m m m m I [ m The we have amey ] ow i the case we have m m m m m That is O the other had i the case we have m m m That is Thus m m [ ] [ ] I I That is U I Hece we have I ( I C U Therefore we have D C Coversey we suppose that I ( I C amey I for a ad a uique Let { y } be the U sequece of {0 } defied by { 0 if is a odd umber y if is a odd umber We put y y m The it foows that y I for a amey we have ad y beog to the same I Thus
It J Pure App Sci Techo 6( (0 7-79 7 y for a Here y y D Thus we have C D Cosequety we have C D m Lemma The Cator set C is totay discoected Proof: Suppose C cotais a iterva (a b The 0 b a Sice im 0 iterva Therefore C is totay discoected Lemma The set D has the properties: (i The epressio uique (ii D has o isoated poit ( for a ad ad a b I we have a b This is a cotradictio Thus C has o f ([ c d] [ c d] φ i D is Proof: (i Suppose that y where m y y m ad y for - ad y We ca assume that ad y 0 The y y m m m m m m y m y 0 is a cotradictio Therefore the epressio y m m m y y y m y m m > 0 m i D is uique This (ii Let D where m {0} ad ε > 0 We show that there eists y( D such that y < ε There eists atura umber such that < ε We put y y m D such that y for ad y The y y Thus y < ε 0 if if 0 By the property (i We have y Thus U ( ε D { } Hece we have that is ot a isoated poit of D Propositio 4 The set C D is a Cator set Proof: By Lemma C D by Lemma C is totay discoected ad by Lemma (ii D has o isoated poit Hece the set C D is a Cator set Lemma 5 Let ( X d ad ( X d be two compact metric spaces ad f : ( X d ( X d be a map If there eists M > 0 such that d f ( f ( y Md ( the f is cotiuous ( y Proof: LetU be a ope set i X We show that f ( U is ope i X Let f ( U that is f ( U SiceU is ope there eists ε > 0 such that f ( U ( f ( ε U ε Let V { y X : d( y < } The V ad V f ( U Thus f ( U is ope i X M
It J Pure App Sci Techo 6( (0 7-79 74 Lemma 6 Let X ad Y be compact Hausdorff spaces respectivey If f : X Y is a bijective ad cotiuous map the (i f is a homeomorphism ad (ii Y is compact space Proof: (i We show that f is cotiuous LetU be a ope set i X We put B X U The B is cosed set Hece B is compact i X Thus f (B is compact i Y amey f ( B f ( X U f ( X f ( U is cosed i Y Thus ( f ( U f ( U is ope i Y Cosequety f is a homeomorphism (ii By assumptio X is compact Sice X ady are homeomorphisms thus we havey is compact Propositio 7 Σ ad D are homeomorphic Proof: Let f : D be the map defied by y f ( where Σ ( y y D ad y We show that f is a homeomorphism of Σ oto D Oe-to-oe: Suppose f ( f ( where ( the y y Thus y y impies for a i The ( Sice the epressio of a eemet i D is uique we Therefore f is oto Cotiuity: We have f ( f ( y y 4 y y By Lemma 5 f is cotiuous Therefore by Lemma 6(i f : Σ D is a homeomorphism amey Σ ad D are homeomorphic Propositio 8 ad are homeomorphic for 46 Proof: We first ote that the map ϕ : Z defied by ϕ ( 5 for f : be the map defied by y f ( is both oe-to-oe ad oto Let where ( Z y y Z y f : is a homomorphism ( ad ϕ ( for a We show that have ( Thus Therefore f is oe-tooe y Oto: Let y y D For this we put ( where The f ( y Oe-tooe:Suppose f ( f ( y Let z f ( f ( y where ( Z y ( y Z z ( z ad z ϕ ( yϕ ( for a Thus ϕ ( yϕ ( for a
It J Pure App Sci Techo 6( (0 7-79 75 Sice ϕ is oto (ie ϕ ( Z we have y for a Z Hece y Therefore f is oe-to-oe Oto: Sice ϕ oe-to-oe there eists the iverse ϕ : Z Let y ( y Z be give For this y we put ( Z where y ϕ ( The f ( y Therefore f is oto Cotiuity: et ( Z The by the defiitio of f we have f ( y ( y Z ad y ϕ ( ϕ ( for a ow d ( f ( f ( d ( y y y y ϕ ( ϕ ( ( ( ( ϕ ϕ ϕ ϕ ( 0 0 0 d ( 0 0 f : is a By Lemma 5 f is cotiuous Therefore by Lemma 6(i homomorphism amey ad are homomorphic Theorem 9 Let X be a compact subspace of R The the foowig are equivaet: ( X is a Cator set C ( X is homeomorphic to C ( X is homeomorphic to D (4 X is homeomorphic to (5 X is homeomorphic to Proof: [( (4] Sice X is cosed R-X is ope Put a mi X b ma X The ( R X ( a b is ope set ad we have: ( R X ( a b J J ( a b The sequece { J } ca be arraged as { } ( { K w w w : wi {0 } for i }( K J whose positio is as foows: (a K : w {0 } for i } are arraged i order of the umbers { w w w i { w w w w : {0} w i for i } (b K w 0( w w resp K ww w eist at the eft (resp right side of K w w w For X we defied a sequece f ( i as foows:
It J Pure App Sci Techo 6( (0 7-79 76 0 if eists at the eft (resp right side of After we defie et 0 (resp if eists at the eft (resp right side of K The f : is a homeomorphism which we show i the foowig Oe-to-oe: We suppose that < y Sice is totay discoected the cosed iterva [ y] cotaied i X Thus there eist ( a b K such that < a < b y J w w w < Hece 0 ad y Thus we have f ( ( ( y f ( y Therefore f is oe to oe Oto: Let w ( w w w For ( w w w et { L } be the sequece of cosed iterva such that L is the argest cosed iterva i X K at the w -had of K ( w is at eft if w 0 ad w is at right if w Suppose that L L are determied The L is the argest cosed iterva i the cosed set ( X K w w w L at the w - had side of K w w w Thus we have L is a o-empty set by the compactess of X Moreover L cosist of sige poit which we show ow Let be a eemet of L The f ( w By (a is a uique eemet of I L Thus f is oto Cotiuity: We show that for ay i X ad for ayε > 0 There eists δ > 0such that if y < δ the f ( f ( y < ε Let X ad ε > 0 are give The there eists > 0 such that < ε Let f ( ad { L} be the sequece defied above The ( L for a We put δ mi{ d( L : } If y < δ the y L for The d ( y y < ε Therefore f is cotiuous By Lemma 6(i f is a homeomorphism Therefore X is homeomorphic to Σ ow usig Propositios 4 8 ad 7 we get ( ( (4 (5 ad (4 ( respectivey Chaotic properties of compact subspaces of the rea ie As we ow chaotic maps are cosidered as those f 's which have the foowig property (cf [4] (C-The set of a periodic poits for f are dese (C- f is trasitive (C- f has sesitive depedece o iitia coditios Those properties are cocered with the orbit of give iitia poit (cf [5] We restrict our attetio to cotiuous maps o compact subspaces of the rea ie ad there eist a ot of chaotic maps such as tet map of the uit iterva ad the two-sided shift map of the Cator sets It is easy to show that tet map is ot a homeomorphism of the uit iterva oto itsef but the two-sided shift map is a homeomorphism of the Cator set oto itsef I [] it was show that o homeomorphisms of the uit iterva are chaotic Usig these
It J Pure App Sci Techo 6( (0 7-79 77 resuts we show that there eists a chaotic homeomorphism of X oto itsef if ad oy if X is homeomorphic to the Cator set Lemma Let f :[0] [0] be a cotiuous map If f is a homeomorphism the f is ot oe-sided topoogicay trasitive Proof: Suppose that f is a homeomorphism The f is mootoicay icreasig or mootoicay decreasig For each case we show that f is ot oe sided topoogicay trasitive First we cosider the case of mootoicay icreasig i which f ( 0 0 ad f ( Let 0be a poit i the ope iterva ( 0 The we have the foowig two cases Case (- [ f ( ] Let ( 0 0 0 U 0 ad V ( 0 The we have f ( U V U V φ ( Case (- [ f ( 0 0 ] Let a mi{ ( 0 f 0} b ma{ 0 f ( 0} U ( a b ad V f (U The we have f ( U V U V φ ( et we cosider the case of mootoicay decreasig i which f ( 0 ad f ( 0 Let y 0 be a poit o the ope iterva ( 0 such that f ( 0 0 The we have the foowig two cases too Case (- [ f ( y0 y0 ] Let U ( 0 y0 ( f ( y0 ad V ( y0 f ( y0 The we have f ( U V U V φ ( Case (-[ f ( y0 y0 ] Let a mi{ y0 f ( y0 } b ma{ y0 f ( y0} U ( a b ad V f (U The we have f ( U V φ ( Lemma Let X be compact subspace of R which has a o-empty ope iterva ( a b ad f : X X a cotiuous map If f is a homeomorphism the f is ot oe sided topoogicay trasitive Proof: Suppose that f is a homeomorphism ad et ( c d be the argest ope iterva i X such that ( c d cotais ( a b Sice X is cosed i R the cosed iterva [ c d ] is cotaied i X I the case where f ([ c d] [ c d] φ ( triviay f is ot oe sided topoogicay trasitive ow we cosider the case where f ([ c d] [ c d] φ for some Let be the smaest positive iteger such that f ([ c d] [ c d] φ Sice [ c d ] is a coected compoet i X ad f is a homeomorphism of X oto itsef the set f ([ c d] is aso a coected compoet i X Thus we have f ([ c d] [ c d] ad the restrictio of f to [ d ] c becomes a homeomorphism of [ c d ] Hece by Lemma f is ot oe sided topoogicay trasitive o [c d] that is there eist two o-empty ope itervas U ad V i [ c d ] such that f ( U V φ for a By cosiderig ope itervas i U adv we ca taeu adv as o-empty ope sets i X ad we have
It J Pure App Sci Techo 6( (0 7-79 78 i f ( U V f i ([ c d] [ c d] φ for i Therefore f is ot a oe sided topoogicay trasitive Lemma Let X be a compact subspace of R which has a isoated poit ad f : X X be a cotiuous map The f does ot have sesitive depedece o iitia coditios Proof: Let a isoated poit The there eistε > 0 such that { y X : d( y < ε } X { } Hece for thisε the coditio d ( y < ε impies y Thus we have d( f ( f ( y 0 for a Lemma 4 The two sided shift map S is chaotic homeomorphism Proof: It is easy to show that S is a homeomorphism ad satisfies three coditios by R L Devaey Cosequety S is chaotic homeomorphism Combiig Lemmas ad 4 we get the foowig propositio Propositio 5 For a compact subspace X of R there eists a homeomorphism of X which satisfies the first two coditios of Devaey if ad oy if X is homemorphic to the Cator set C Moreover Lemma 4 meas that this is equivaet to that there eists a chaotic homeomorphism of X amey we have the foowig theorem Theorem 6 Let X be a compact subspace of R The there eists a chaotic homeomorphism f : X X if ad oy if X is homeomorphic to a Cator set Proof: We show the foowig two facts: (i If X is homeomorphic to the there eists chaotic homeomorphism f : X X (ii If there eist s f : X X is a chaotic homeomorphism the X is homeomorphic to To prove (i suppose h : X is a homeomorphism By Lemma 4 S : is a chaotic homeomorphism The we have the foowig commutative diagram f X X h The f h o So h Hece f is a chaotic homeomorphism (iiwe show it by cotrapositio Suppose X is ot homeomorphic to the by Lemmas ad 4(ii we have (a X is ot totay discoected or (b X has a isoated poit I that case (a if X is ot totay discoected the there eist X such that C ( { } The there eists y such that f C( If < y the [ y] C( ad if > y the [ y] C( ie there eists ( a( [ a b] C( X S h
It J Pure App Sci Techo 6( (0 7-79 79 The by Lemma there is o homeomorphism which is chaotic I the case (b if X has a isoated poit the by Lemma there is o homeomorphism which is chaotic Remar 7 We ote that i the case where X is a fiite subset of R there eists a homeomorphism of X which satisfies the first two chaotic coditios but does ot satisfy the third coditios by Devaey I Lemma 4 we have show that there eists at east oe chaotic homeomorphism of the Cator set oto itsef that is the two sided shift map I [] it was show that this chaotic map is a map i a famiy of chaotic homeomorphism Refereces [] J Bas J Broos G Cairs G Davis ad P Stacey O devaey's defiitio of chaos Amer Math Mothy 99 (99 4 [] P Ahmed ad S Kawamura Chaotic homeomorphisms of the compact subspace of rea ie Bu of Yamagata Uiversity Japa at Sci6 (4 008 [] P Ahmed ad S Isam Chaotic behavior i dyamica systems of homeomorphism o uit iterva Joura of Bagadesh Academy of Scieces ( (008-9 [4] R L Devaey A Itroductio to Chaotic Dyamica Systems d Editio Addiso- Wesey: Redwood City 989 [5] S M Uam ad J Vo euma O combiatio of stochastic ad determiistic processes Preimiary report Bu Amer Math Soc 5(947 0