International Journal of Pure and Applied Sciences and Technology

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1 It J Pure Appl Sc Techol, () (00), pp Iteratoal Joural of Pure ad Appled Scece ad Techology ISSN Avalable ole at wwwjopaaat Reearch Paper Some Stroger Chaotc Feature of the Geeralzed Shft Map Idral Bhaum,* ad Baya S Choudhury, Departmet of Mathematc, Begal Egeerg ad Scece Uverty, Shbpur, Howrah - 703, Wet Begal, Ida * Correpodg author, e-mal: (raadra006@yahooco) (Receved: --00 ; Accepted: 7--00) Abtract: Recetly, we have troduced the oto of the geeralzato of the hft map, that, the geeralzed hft map, the ymbol pace Σ I th paper we have proved ome troger chaotc properte of the geeralzed hft map A example ha bee gve the lat ecto Keyword: Chao, Symbol pace, Geeralzed hft map, Topologcally mxg, Chaotc depedece o tal codto Itroducto A dyamcal ytem a tudy of how phycal ad mathematcal ytem evolve wth tme, developed through the collectve effort of mathematca ad cett may deferet feld A dyamcal ytem clude the followg compoet: a phae pace S whoe elemet repreet poble tate of the ytem; tme t (whch may be dcrete or cotuou) ad a evoluto law (that, a rule that allow determato of the tate at tme t from the owledge of the tate at all prevou tme) Hece a geeral dyamcal

2 It J Pure Appl Sc Techol, () (00), ytem ca be defed a a par ( X, f ) cotg of a et X together wth a cotuou map f from X to telf Chaotcty a mportat property for ay dyamcal ytem The tudy of chaotc dyamc ha become creagly popular at the preet day Although there ha bee o uverally accepted mathematcal defto of chao, t geerally beleved that f for ay ytem the dtace betwee the earby pot creae ad the dtace betwee the far away pot decreae wth tme, the ytem ad to be chaotc Hece a dyamcal ytem chaotc f the orbt of t (or a ubet of t) are cofed to a bouded rego, but tll behave upredctably Th prt caught everal equvalet defto [6, 7, 8, 9, 0, ] of chao The term chao wa frt ued mathematcally by L ad Yor ther paper Perod three mple chao [9] 975 Symbolc dyamcal ytem ( Σ, σ ) ad ( Σ, σ ), where Σ the equece pace, σ the hft map ad σ the geeralzed hft map, are alo example of chaotc dyamcal ytem I partcular there are everal wor o ymbolc dyamc where dyamc are repreeted by map o ymbol pace Some of thee wor are oted referece [,, 3, 4, 7, 8,, ] Of partcular teret the pace Σ whch ha bee codered a large umber of wor, where = α : α = ( α α ), α 0 or }, a metrc pace { 0 = wth the metrc (, ) = t d t, where = ( + ) = 0 0 ad t = ( t t 0 ) are two pot of Σ It eay to prove that, by our choe metrc the maxmum dtace betwee ay two pot of Σ Recetly, the preet author troduced the cocept of geeralzed hft map [] I th paper we have proved ome troger chaotc properte of the geeralzed hft map I Theorem 3, t proved that the geeralzed hft map topologcally mxg o Σ The we have proved, Theorem 3, that the geeralzed hft map ha chaotc depedece o tal codto We alo have gve a example of a cotuou fucto whch topologcally tratve but ot chaotc the ee of Du [8] Mathematcal Prelmare Here we gve ome defto ad lemma whch are requred for ext two ecto

3 It J Pure Appl Sc Techol, () (00), Defto (Shft map [7]) The hft map σ : Σ Σ defed by σ ( α) = ( α α ), where α = ( α 0 α) ay pot of Σ Defto (Geeralzed hft map []) The geeralzed hft map σ : Σ Σ defed by σ ( ) = ( + ), where = ( 0 ) ay pot of Σ ad ay teger Defto 3 (Topologcally tratve [7]) Let ( X, ρ) be a compact metrc pace A mappg f : X X ad to be topologcally tratve f for ay par of o -empty ope et K, L X there ext 0 uch that f ( K) L φ Defto 4 (Topologcally mxg []) Let ( X, ρ) be a compact metrc pace ad f : X X be a cotuou map The map f called topologcally mxg f for ay two o-empty ope et U, V X there ext m 0 uch that for all m, f ( U ) V φ Defto 5 (Setve depedece o tal codto [7]) Let ( X, ρ) be a compact metrc pace A cotuou map f : X X ha etve depedece o tal codto f there ext δ > 0 uch that, for ay x S ad ay eghborhood N (x) of x there ext y N(x) ad 0 uch that ρ ( f ( x), f ( y)) > δ Defto 6 (L -Yore par [5]) A par (, ) X x y called a L -Yore par (wth modulu δ ) f p p Lt Sup ρ( f ( x), f ( y)) δ p p p ad Lt If ρ ( f ( x), f ( y)) = 0, p where ( X, f ) a dyamcal ytem, X beg a compact metrc pace wth the metrc ρ ad f a cotuou mappg o X Defto 7 (Chaotc depedece o tal codto [5]) A dyamcal ytem ( X, f ) ha chaotc depedece o tal codto f for ay x X ad ay eghborhood N (x) of x there y N(x) uch that the par (, ) X x y L -Yore We alo eed the followg lemma

4 It J Pure Appl Sc Techol, () (00), Lemma [7]: Let, t Σ ad = t, for = 0,,, m The d(, t) < m ad coverely f d(, t) < the m = t, for = 0,,, m 3 The Ma Theorem Theorem 3 The geeralzed hft map σ : Σ Σ topologcally mxg o Σ Proof: We tae ay two o -empty ope et U ad V of Σ Let be ay pot uch that m { ( u, β ) } = ε u = ( u u ) U d, for ay β belog to the boudary of the et U 0 ad v = ( v v ) V m β = ε d ( v, ), for ay 0 be ay pot uch that { } β belog to the boudary of the et V, where ε, ε > 0 We ow chooe two potve teger ad <ε uch that ad <ε Latly, we coder the equece of pot gve ( ) by, α u u u (0) v v v ), for =,3, = ( 0 0 ad α u u u v v v ) = ( 0 0 Now, d ( u, α ) < < ε, =,,, by Lemma (3) Hece α U, =,,, that, σ ( α ) σ ( U ), for ay 0 O the other had, σ ( α ) = ( v0v v ) Hece d ( σ ( α), v) < < ε, by applyg Lemma aga (3) Th gve σ ( α ) V

5 It J Pure Appl Sc Techol, () (00), I vrtue of (3) ad (3) we ca ay that σ ( U ) V φ Next we coder the pot + α The σ α ) ( v v v ) Whch aga belog to V Hece ( = 0 + σ ( U ) V φ Cotug th proce by tag allα we get σ ( U V φ, for ) all Hece σ topologcally mxg o Σ Theorem 3 The dyamcal ytem ( Σ,σ ) ha chaotc depedece o tal codto Proof: At frt we gve ome otato whch help u to prove Theorem 3 Let = ( 0 ) be ay pot of Σ ad U be ay ope eghborhood of Let S = 0 ad P = p p 0 pm be two fte equece of 0 ad, the S P = p Further, f we uppoe that T, 0 p0 p, T p are p fte equece of 0 ad ; T T Tp maer a above 3 If β ay bary umeral, we deote the complemet of β = 0 or, the β = or 0 m, T ca be defed a mlar β by 4 Let F, 0) = ( ) ( F ( ) (, ) + = β That, f,, ad o o Note that for ay eve teger m, F (, + m) a fte trg of legth ( + m) 5 Latly, we tae t Σ uch that ( (0) () F (, + 0) F (, + ) F (, 4) ) t = 0 +, where ( α ) = αα α tme We coder the pot ad the ope eghborhood U of defed the above otato Sce U ope we ca alway chooe a > 0 ε, uch that m{ d (, α )} = ε, for ay

6 It J Pure Appl Sc Techol, () (00), α belog to the boudary of the et U We chooe o large that < ε By our cotructo ad t agree up to Hece d (, t) < < ε, by Lemma So 3 σ ad t) ( ) 3 t U Now ) ( ) ( = σ ( = Note that t cot of ftely may fte equece of the type A (, + m) So we get Lt Sup d( σ ( ), ( )) (( 3 4 ), ( 3 σ t Lt d 4 )) Lt ( ) = (33) Hece, Lt Sup d( σ ( ), σ ( t)) = 4 Smlarly, ) ( ) Aga we get that ( = σ ad σ t) ( ) ( = (( ), ( )) Lt If d( σ ( ), σ ( t)) Lt Lt ( ) = 0 (34) Hece, Lt If d( σ ( ), σ ( t)) = 0 From (33) ad (34) t proved that the par (, t) L -Yore Hece the dyamcal ytem ( Σ,σ ) ha chaotc depedece o tal codto 4 Cocluo I th paper we have proved ome troger chaotc properte of the geeralzed hft map Alo the property Defto 6 very mportat for ay dyamcal ytem, becaue th property maly baed o L -Yore par but ha ome commo feature of etve depedece o tal codto Hece we ca ay that the geeralzed hft map ha a property whch baed o L -Yore par but have ome commo feature of etve depedece o tal codto Alo we have proved that the geeralzed hft

7 It J Pure Appl Sc Techol, () (00), map topologcally mxg o Σ, whch a property troger tha topologcal tratvty Latly, we gve a example of a cotuou fucto whch topologcally tratve but ot chaotc the ee of Du [8] Example 4: Let f :[,] [,] be a fucto defed by f ( x) = 7 7 x +, x x, x 0 7 x, 0 x The fucto defed above obvouly a cotuou fucto It ca be ealy proved that the fucto topologcally tratve But t ot chaotc the ee of Du ce the perod two pot cloe to each other 7 ad the cloed terval [ 0,] are jumpg alteratvely ad ever get 3 Acowledgemet Idral Bhaum acowledge h father Mr Sadha Chadra Bhaum for h help preparg the maucrpt Referece [] I Bhaum ad B S Choudhury, Dyamc of the geeralzed hft map, Bull Cal Math Soc, 0(5) (009), [] I Bhaum ad B S Choudhury, The hft map ad the ymbolc dyamc ad applcato of topologcal cojugacy, J Phy Sc, 3 (009), [3] I Bhaum ad B S Choudhury, Topologcally cojugate map ad ω -chao ymbol pace, It J Appl Math, 3() (00), [4] I Bhaum ad B S Choudhury, Some uow properte of ymbolc dyamc, It J Appl Math, 3(5) (00),

8 It J Pure Appl Sc Techol, () (00), [5] F Blachard, E Glaer, S Kolyada ad A Maa, O L -Yore par, J Ree Agew Math, 547 (00), 5-68 [6] L S Bloc ad W A Copple, Dyamc Oe Dmeo, Sprger Lecture Note, 53, Sprger-Verlag, Berl, 99 [7] R L Devaey, A troducto to chaotc dyamcal ytem, d edto, Addo - Weley, Redwood Cty, CA, 989 [8] B S Du, O the ature of chao, arxv:mathds , February 006 [9] T Y L ad J A Yore, Perod three mple chao, Amer Math Mo, 8(0) (975), [0] S L, ω -chao ad topologcal etropy, Tra Amer Math Soc, 339() (993), [] W Parry, Symbolc dyamc ad traformato of the ut terval, Tra Amer Math Soc, () (966), [] C Robo, Dyamcal ytem: Stablty, ymbolc dyamc ad chao, d edto, CRC pre, Boca Rato, FL, 999