Dyamics of Piecewise Cotiuous Fuctios Sauleh Ahmad Siddiqui April 30 th, 2007 Abstract I our paper, we explore the chaotic behavior of a class of piecewise cotiuous fuctios defied o a iterval X i the real lie. A fuctio or curve is piecewise cotiuous if it is cotiuous at all but a fiite umber of poits ad the left ad right had side limits exist at those poits. I particular we will be lookig at the piecewise cotiuous fuctio f : X " X for which the pieces are affie. We will show that show that f is locally evetually oto, has dese periodic poits, ad has sesitive depedece o iitial coditios. We also show that f is quasicojugate to a shift map of fiite type to prove that f is chaotic. While fuctios such as f are kow to be chaotic through proofs developed by measure theory, our paper is uique i that we use topological dyamics to show f is chaotic. Future work may exted these results to other compact metric spaces ad more geeral piecewise cotiuous fuctios. 1
\Sectio 1. Itroductio A dyamical system is a meas of describig how oe state develops ito aother state over the course of time. Techically, a dyamical system is a smooth actio of the real umbers or the itegers o aother object [W]. The study of dyamical systems, the brach of mathematics that attempts to uderstad processes i motio, traces its roots to the developmet of models based o oliear ordiary differetial equatios ad their solutios [D]. Newto worked with such problems, but Poicaré was the first mathematicia to study dyamical systems i depth; he came up with a idea similar to chaos whe he submitted his attempted solutio to the -body problem about a hudred years ago. After that other mathematicias, icludig Stephe Smale who used Poicaré s techiques to get ew results, studied this pheomeo of chaos (though it was t called chaos just yet!) i the 1960 s. Oe mathematicia foud chaos by accidet. I 1961, Edward Lorez was modelig chagig weather patters whe he chaged the iitial iput coditios by a extremely tiy fractio. To his amazemet, his predicted weather patters chaged drastically after oly a short period of time [D]. This observatio of a sesitive depedece o iitial coditios would later form the basis of chaos theory. Fially, i 1975 James Yorke ad T.-Y. Li published a paper titled Period Three Implies Chaos (Iterestigly, their results were first aticipated by Sarkovskii i the 1964). This was the first time the word chaos was used i scietific literature to describe the pheomea discovered by Poicaré i the late 19 th cetury [D]. Although Yorke ad Li coied the term more tha thirty years ago, mathematicias have still ot settled o a uiform defiitio of chaos [D]. Robert Devaey s defiitio is the most widely used today ad states that a dyamical system F : X " X is chaotic if the followig three coditios hold [D]. i) F is trasitive. A dyamical system is trasitive if for ay two o-empty ope sets U,V " X, there exists a k such that F k (U) "V # $. ii) F has sesitive depedece o iitial coditios. A dyamical system has sesitive depedece o iitial coditios if there exists a sesitivity costat " F > 0 such that for all o-empty ope sets U " X, ad for ay two distict poits x ad y i U, there exists a k such that F k (x) " F k (y) > # F. 2
iii) Periodic poits of F are dese i X. A poit x " X is a periodic poit of F if there exists a k > 0 such that F k (x) = x. The set of periodic poits of F is dese i X if for all " > 0, ad for all x " X, there exists a periodic poit p such that x " p < #. Please ote that for all x " X, F 0 (x) = x, F 1 (x) = F(x), ad F k (x) = F " F k#1 (x) for all k " #. Mathematicias have devoted a lot of time towards studyig cotiuous dyamical systems because they ca be used to model several atural pheomea (such as weather patters) as well as solve complex theoretical problems (such as the -body problem). Dr. Aalisa Craell has studied quasicotiuous dyamical systems, which was the origial motivatio for this project [C1]. Accordig to Dr. Mario Martelli, quasicotiuous systems serve to bridge the gap i the study of dyamical systems betwee measure theory ad topological dyamics [M]. Dr. Youga Choi at Motclair State Uiversity, a measure theorist, curretly studies piecewise expadig maps [C2, C3]. Our project is based o Choi s work, though we will be takig the topological dyamics route istead of usig measure theory. It is for this reaso that while we kow about her work ad results, we have ot read her methods or techiques. This paper will show that our fuctio f, similar to Choi s piecewise expadig map, is chaotic. The ext sectio will describe our fuctio f. After that, we will look at the properties of ", a well-kow shift map of fiite type, ad our fial sectio will focus o showig that the system f : X " X is quasicojugate to " : # $ #. Sectio 2. Uderstadig f Whe readig this sectio, it will be helpful to look at Figure 2.1 o page 5. We start by defiig the space X, o which our fuctio f operates. We defie X " # to be a closed iterval such that X = U I ad I k k 's are half-ope itervals for all 1 " k " ( #1) ad closed for k =. k=1 Specifically, we defie X = [d 0,d +1 ], I k = [d k"1,d k ) for all 1 " k " ( #1), I = [d,d "1 +1 ], ad we defie the poit of discotiuity i I as d. While we are allowed to specify d 0,d 1,d 2,Kd "1 ad d +1, d is determied by the legths of I 1 ad I. We deote the legth of a iterval I k as " k. Note that " k = d k # d k#1 for all 1" k " ( #1), ad " = d +1 # d #1. I Figure 2.1 = 4; that is, X is divided ito four itervals. 3
We defie f : X " X i the followig fashio. The fuctio f is a piecewise affie, piecewise cotiuous fuctio such that f : X " X where f I k is cotiuous for all "1 1" k " ( #1), f is cotiuous o U I k, ad f has two poits of discotiuity; oe at d +1 ad oe k=1 at d. Also, f (I k ) = I k +1 for all 1" k " ( #1), ad f (I ) = I 1 U I. Furthermore, f '(x) is costat ad positive o the iterior of each I k. For 0 " i " # 2, f (d i ) = d i+1 ; f (d "1 ) = d 0 ; f (d ) = d "1 ; ad f (d +1 ) = d +1. Agai, Figure 1 gives a picture of f for = 4. The algebraic equatio for the fuctio looks like the followig: + $ x + d 1 # " ' 2 - & d 0 ) for d 0 * x < d 1 - % ( - " 3 $ x + d 2 # " ' 3 - & d 1 ) for d 1 * x < d 2 - % ( f (x) =, M - " + $ x + d 0 # " + " ' 1 - & d #1 ) for d #1 * x < d - " % " ( - " + $ x + d #1 # " + " ' 1 - & d ) for d * x * d +1. " % " ( Note that the slope of the lies i the last iterval (for d "1 # x # d +1 ) is greater tha 1, which meas that ay two poits close by i I = [d "1,d +1 ] will be pushed far apart by oe iteratio of f. I Figures 2.1, 2.2, ad 2.3, we look at images of f uder three successive iterates. The graphs below are of the simples case where all the itervals are of equal legth ( = = " 3 = " 4 ) ad the umber of itervals i our partitio = 4. Equal slopes are colored with the same color. Also see below, multiple iteratios of f icrease the slopes of lies i the fourth iterval. 4
f (X) I 1 I 2 I 3 I 4 X Figure 2.1: The graph of f : X " X. 5
f 2 (X) I 1 I 2 I 3 I 4 X Figure 2.2: The graph of f 2 : X " X. 6
f 3 (X) I 1 I 2 I 3 I 4 X Figure 2.2: The graph of f 3 : X " X. 7
We may guess from the graphs that successive iterates of f will evetually map ay iterval I k " X across all of X. We ow show this result more rigorously i Theorem 2.1. Theorem 2.1. For every k such that 1" k ", there exists a i " # + such that f i (I k ) = X. Proof. Pick k such that 1 " k ". We begi by cosiderig the last (th) iterval. If k =, cosider f "1 (I ). The f "1 (I ) = f "2 (I 1 ) # f "2 (I ) ( ) = f "3 (I 2 ) # f "3 (I 1 ) # f "3 (I ) ad f "1 (I ) = I "1 # I "2 #$$$# I 2 # ( I 1 # I ) = X. = f 1 (I "2 ) # f 1 (I "3 ) #$ $ $# f 1 (I 1 ) # f 1 (I ), If 1 " k " #1, we kow that f "k (I k ) = I, ad therefore ad we are doe. f 2"1"k (I k ) = f ("1)+("k ) (I k ) = f ("1) f "k (I k ) = f "1 (I ) = X, The result of Theorem 2.1 is our first step to showig that f is locally evetually oto (defied later i Defiitio 3.1). While we will later show that f is locally evetually oto by establishig a quasicojugacy, Theorem 2.1 is a ice result that does t ivolve complicated methods. We have metioed how icreasig slopes of the successive iterates hit at the fuctio f beig chaotic. The followig two theorems show that f has sesitive depedece o iitial coditios. But to prove theorem 2.2, we eed to prove the followig lemma. Lemma 2.1. Pick ay two poits a ad b i the th iterval I. The, if the distace betwee a # 2 " ad b is greater tha max 1 " " & $, ', f (a) ad f (b) will be i differet itervals. % + " + " ( 8
Proof. Recall that = I 1 = d 1 # d 0 ad " = I = d +1 # d #1. Sice the slope of f i the th iterval is give by + ", we kow that + " " = 1. Therefore, the distace betwee d "1 ad " " d # d #1 d is give by " + " ad, similarly, the distace betwee d ad d +1 is give by " 2 + ". Therefore, if a ad b are i the th iterval ad the distace betwee them is greater tha # 2 " max 1 " " & $, ' % + " + " (, f (a) ad f (b) have to be i differet itervals (oe i I ad oe i I ). 1 This becomes clearer with the picture below. Figure 2.2: Behavior of fuctio i I 9
Now we are ready to prove the followig theorem. Theorem 2.2. For ay two poits a ad b i X, there exists a i " 0 such that f i (a) ad f i (b) are i separate itervals. Proof. If a ad b are i separate itervals the f 0 (a) ad f 0 (b) are i separate itervals so we are doe. Assume istead that a ad b are i the same iterval I k with legth " k ad there exists a M " N such that for all m < M, f m (a) ad f m (b) are i the same iterval. By usig the costat slope we get f (b) " f (a) = # k +1 # k b " a, f 2 (b) " f 2 (a) = # k +2 f (b) " f (a) = # $ k +2 # k +1 ' & b " a) # k +1 # k +1 % # k ( = # k +2 # k b " a, ad so o. The distace betwee the ( k)th images of a ad b i I is give by f "k (b) " f "k (a). We see the that f "k (b) " f "k (a) # $ $ k b " a. By Lemma 2.1, if the distace # 2 " betwee two poits is greater tha max 1 " " & $, ' i the th iterval, the the ext iteratio % + " + " ( of f will take the poits ito separate itervals. There are three possibilities at this poit. Oe possibility is that f "k (b) " f "k (a) # $ % 2 $ b " a # max 1 $ $ ( &, ) $ k ' + $ + $ * i which case the ext iteratio of f takes images of a ad b ito differet itervals ad we are doe. Aother possibility is that the ext iteratio of f takes these images ito the 1 st iterval ad the subsequetly back ito the th iterval after aother 1 iteratios. Oce the images are back i the th iterval, they ca oce agai either go ito the 1 st iterval or the th iterval. Fially, the ext iteratio of f ca take images of a ad b back ito the th iterval. Let " = "(M) deote the umber of times f takes images of a ad b to the first iterval durig the first M 10
iteratios. Let " = "(M) deote the umber of times f takes images of a ad b to the th iterval durig the first M iteratios. We claim that if M = " k + # + $ the f M (b) " f M (a) = # $ b " a # ' & +1) # k % # 1 ( * $ #1 + ' & +1). % # ( Why is this true? We kow that if we pick ay two poits c ad e i the th iterval I, if f (c) ad f (e) are i the 1 st iterval, the distace betwee them is give by f (c) " f (e) = # 1 + # # c " e. Subsequetly, after ( 1) further iteratios of f, the images of c ad e arrive back to I, where $ their distace is give by f (c) " f (e) = c " e # ' & +1). Similarly, the distace betwee ay two % # 1 ( poits c ad e i the th iterval I, if f (c) ad f (e) are i the th iterval, is give by f (c) " f (e) = # 1 + # # c " e. The, we see that f "k +# +$ (c) " f "k +# +$ (e) % & c " e & # $ ' * ' &1 + & ) +1 *, ),. & k ( & 1 + ( + We will ow use the above claim to show that f i (a) ad f i (b) are i separate itervals for # " some i. Sice both & # " % +1( ad 1 & % +1( are greater tha oe, " $ b # a " ' & +1) $ ' $ " ' " k % ( & * $ "1 ' & +1) % " ( evetually be greater tha ay positive value as either " or " get large. Thus, for large eough values of m, f m (b) " f m (a) # $ % b " a $ ( ' +1* $ k & ) % ( ' +1* & $ ) + $1 m"(+ +1)+k # $ + $ ad we have foud a i for which f i (a) ad f i (b) are i separate itervals. + ca Theorem 2.3. The fuctio f : X " X has sesitive depedece o iitial coditios. Proof. We wat to show that there exists a sesitivity costat " f > 0 such that for ay o-empty ope set U " X, there exist two distict poits a ad b i U for which there exists a k such that f k (a) " f k (b) > # f. We will show that " f = mi {# 2,# 3,...# ($1) } is a sesitivity costat. Pick 11
a ope set U " X ad choose two distict poits a ad b i U where a < b such that a ad b are i the same iterval; the by Theorem 2.2, there exists a smallest k such that f k (a) ad f k (b) are i separate itervals. The oly way f k (a) ad f k (b) will be i separate itervals is that if f k"1 (a) ad f k"1 (b) are i the th iterval. The ext iteratio of f will esure that f k (a) ad f k (b) are i separate itervals (oe of them will be i the first iterval ad the other oe will be i the th iterval). The f k (a) " f k (b) > (# 2 + # 3 + # 4 + $$$ + # "1 ) > % F. Thus, f has sesitive depedece o iitial coditios. To complete the proof that f is chaotic, we eed to show that f is trasitive ad has dese periodic poits. This is difficult to do directly, so we will first study a well-kow chaotic fuctio, the shift map of fiite type, i sectio 3. We will the show that f is chaotic because it is quasicojugate to this particular shift map of fiite type. Sectio 3. Uderstadig σ We chose to study σ, a shift map of fiite type, because the space o which it operates, Σ, gives us a good way to represet poits movig ito differet itervals. That is, Σ codes the iterates of poits i X as they move because of f. We will describe this i more detail below. We defie Σ by " = {# = # 0 # 1 # 2... # j $ { 1,2,...,},%j,# j # j +1 $ A} where A (meaig allowable pairs ) is defied by A = { 12,23,34,...,( "1),,1}. That is, for all 1 " k " ( #1) if " j = k, the " j +1 = k +1; ad if " j =, the " j +1 = or " j +1 =1. Thus, we ll show a poit that starts off i the 2 d iterval ca be coded by " = 2345...( #1)1234... We will see that this correspods to a poit x " I 2 for which ad so o. f (x) " I 3, f 2 (x) " I 4,..., f #2 (x) " I, f "1 (x) # I, f (x) # I, f +1 (x) # I 1... 12
We defie a metric d o Σ to describe the distace betwee two poits " = (" 0...) ad " = (" 0...) as d [",#] = % & j= 0 " j $ # j 2 j. This metric iduces a topology o Σ whose basis is defied as the set of all subsets C m [" 0 ] = {" 0 $ 0 $ 2... " m#1 $ 0 % A} where " 0 $ 0 $ 2... is a allowable sequece i Σ. That is, C m is a set of all elemets for which the first m terms are the same. Without loss of geerality, we may assume that " m#1 =. This is because the oly place two sequeces i Σ ca differ is where the previous positio is occupied by. I symbolic dyamics, these sets are commoly called cyliders. Whe " 0 are uderstood, we will abbreviate C m [" 0 ] to C m. The fuctio " : # $ # is defied, for all " # $ as "(#) = "(# 0 # 1 # 2...) = # 1 # 2... It is well kow that σ, the shift map of fiite type, is cotiuous [D]. To show that σ is chaotic o Σ, we oly eed to show that it is trasitive ad has dese periodic poits [B]. I Theorem 3.1 we will show that σ is locally evetually oto, which is stroger tha showig it is trasitive. Theorem 3.2 shows that σ has dese periodic poits. Defiitio 3.1. A fuctio F : X " X is locally evetually oto if for every o-empty ope set U " X, there exists a i such that F i (U) = X. Lemma 3.1. Give a locally evetually oto fuctio F : X " X it follows that F is trasitive. Proof. Pick o-empty ope sets V,W " X. We kow there exists a i " N such that F i (V ) = X. The, sice W is a o-empty ope subset of X, F i (V ) "W = X "W = W # $ ad we are doe. Theorem 3.1. The shift map of fiite type " : # $ # is locally evetually oto ad thus trasitive. Proof. Pick a ope set U " #. The, there exists a basis elemet C m [" 0 ] such that C m " U. The, " m (C m ) = {" m (#)} = {" m (# 0 # 1 # 2...# m$1 % 0 % 1 % 2...)} = {% 0 % 1 % 2...}. 13
Now, " m#1 $ 0 is a allowable pair, so " 0 is ot arbitrary. Sice all the poits of C m are similar up to " m#1, ad " m#1 =, " 0 must be either 1 or. After aother 1 iteratios, " #1 ca be ay member of a sequece. Sice " #1 " " +1... is ay allowable sequece i Σ, " m +#1 (C m ) is the set of all allowable sequeces i Σ, ad thus the whole space Σ. Sice C m is a subset of U, " m + (U) = # ad we are doe. Theorem 3.2. The shift map of fiite type " : # $ # has dese periodic poits. Proof. Pick a o-empty ope set U " #. The, there exists a basis elemet C m [" 0 ] so that C m is a ope subset of U. The, there exists a q > 0 such that " q # 0 is a allowable pair. Now, pick a elemet " # C m such that " = " 0 $ 0 $ 2...$ q " 0 $ 0 $ 2...$ q...= " 0 $ 0 $ 2...$ q. The, " m +q +1 (#) = " m +q +1 (# 0 # 1 # 2...# m$1 % 0 % 1 % 2...% q ) = (# 0 # 1 # 2...# m$1 % 0 % 1 % 2...% q ) = #. Therefore, is a periodic poit of σ with periodicity m + q + 1 i U. Sice we chose a arbitrary ope set U i Σ, it follows that σ has dese periodic poits. (Please ote that " sigifies a elemet of Σ while " 0 ( ) sigifies repeatig a fiite sequece.) Sectio 4. Quasicojugacy This sectio of the paper is the fial step to prove that f is chaotic. We have already show that f has sesitive depedece o iitial coditios ad we have show that σ is trasitive ad has dese periodic poits. If the dyamical system ( f,x) is quasicojugate to the dyamical system (",#), the f is chaotic [C1]. Before we defie what it meas for a system to be quasicojugate to aother system, we must defie what it meas for a fuctio to be quasicotiuous. Defiitio 4.1. A fuctio F : X " Y, where X ad Y are metric spaces, is quasicotiuous if "x # X ad for "# > 0, there exists a o-empty ope set U " X with x " U ad F(U) " B # (F(x)). 14
I Defiitio 4.2 we will defie a fuctio S betwee the dyamical systems ( f,x) ad (",#). Defiitio 4.3 the describes the ecessary coditios for S to be a quasicojugacy. Defiitio 4.2. We defie a fuctio S : X " # by S(x) = " 0... where f i (x) " I #i ad [" 0...] # $ for all x " X. Defiitio 4.3. Let X ad Y be metric spaces ad let " ad " be fuctios such that " : X # X ad " :Y # Y. Let F be a quasicotiuous fuctio such that F : X " Y, F "1 :Y # X is cotiuous, F : X " Y is oe to oe ad oto (with the exceptio of a coutable umber of poits), ad F o " =# o F. The, the map F is called a quasicojugacy betwee the systems (", X) ad (",Y). Now cosider S : X " # where we claim S is a quasicojugacy betwee the systems (X, f ) ad (",#). Sice we kow that ay two poits i X will ed up i separate itervals by successive iteratios of f, each poit ca be writte as a uique sequece of symbols i the sequece space ". Thus, a poit i the 1 st iterval ca be writte as (123...123...) i the sequece space, depedig o which iterval it goes to from the th iterval. The followig theorem is just a cosequece of the fact that ay two distict poits i X have uique represetatios i the sequece space. Theorem 4.1. S : X " # is oe-to-oe. Proof. We wat to show that for ay two distict poits a ad b i X, S(a) ad S(b) are distict poits i Σ as well. Pick two distict poits a ad b i X. The, by Theorem 2.2, there exists a i " 0 such that f i (a) ad f i (b) are i separate itervals. This meas that the sequeces give by S(a) ad S(b) will evetually be differet at the ith term. Thus, for all a ad b i X, S(a) ad S(b) are distict poits i Σ. We ow show that S : X " # is almost oto (that is, it is oto except for a coutable umber of poits). The reaso that S : X " # is ot completely oto is that sequeces i Σ that start with some other umber " 0 but ed i have o correspodig poits i X. Oly 15
d +1 " X maps oto i Σ but, for example, there is o poit i X that maps oto ( 1) i Σ. Therefore, if we igore all the poits i Σ ot edig i a cotiuous sequece of s, S : X " # is oto. It s clear that the sequeces i Σ that ed with are coutable. Note that the periodic poits described i Theorem 3.2 do t ed i this way, so it turs out that the periodic poits i Σ have couterparts i X ad vice versa. We defie the space Σ without all sequeces that ed i as ". Theorem 4.2. S : X " # is oto. Proof. We wat to show that for all " i ", there exists a x i X such that S(x) = ". Pick " i " such that " = " 0... Now cosider the set N U N = I f "i (I #i ) = I #0 $ f "1 (I #1 ) $ f "2 (I #2 ) $ f "3 (I #3 )...f "N (I #N ). 0 I i= 0 First, ote that U 0 = f "i (I #i ) = I #0 is closed, o-empty, ad cotais i= 0 1 U 1 = I f "i (I #i ) = I #0 $ f "1 (I #1 ). i= 0 Notice that f (U 0 ) = I "1 ad f (U 1 ) = f ( I "0 # f $1 (I "1 )) = f (I "0 ) # f ( f $1 (I "1 )) % I "1. I geeral, f i (U j ) " I #i. We will use iductio to show that for all N, U N is closed, o-empty ad cotais U N +1. k Now, suppose that U k = I f "i (I #i ) is closed ad o-empty. The, cosider k +1 I i= 0 U k +1 = f "i (I #i ) = U k $ f "(k +1) (I #k+1 ). We see that U k +1 is closed because it is the itersectio of i= 0 k +1 two closed sets. Also, U k +1 " U k because U k +1 = I f "i (I #i ) = U k $ f "(k +1) (I #k+1 ). Furthermore, U k +1 is o-empty because the closed iverse images will cotai a set of poits that itersect with the origial iterval I "0. Therefore, by the priciple of mathematical iductio, U N is closed, oempty iterval ad cotais a closed ad o-empty iterval U N +1 for all N. Now let U = I U N. Sice U is composed of a ifiite itersectio of closed bouded itervals, it cotais oe poit. i= 0 " N= 0 16
Let this poit be x. Note that, sice x " U, x " U 0 which meas that x " I #0. I geeral, x " U N for all N. The, f (x) " I #1, f 2 (x) " I #2, ad, i geeral, f N (x) " I #N. Therefore, x ca be represeted by the sequece " = " 0... i ". Therefore, S : X " # is oto. The followig two theorems will complete our proof that S is a quasicojugacy. We will first show that S is quasicotiuous ad the that S -1 is cotiuous. Theorem 4.3. S : X " # is quasicotiuous. Proof. From our defiitio of quasicotiuity, it is sufficiet to show that "x # X ad for "# > 0, "# > 0 such that S ((x, x + ")) # B $ (S(x)). Pick x " X ad choose " > 0. Let S(x) = " 0... Choose m such that 2 m +1 < ". By the defiitio of the metric o Σ, there exists a cylider C m such that C m [" 0 ] $ B % (S(x)). Therefore, the elemets of C m are give by C m = {" 0 $ 0 $ 2...} where " 0 are the first m terms of S(x) ad " m#1 $ 0 $ 2... is a allowable sequece i Σ. Now choose " small eough such that the followig coditios hold. 1) The ope iterval(x, x + ") does ot cotai d k for ay k. 2) For all atural umbers N < m, the ope iterval ( f N (x), f N (x + ")) does ot cotai d k for ay k. 3) For all atural umbers N < m, f N (x + ") # f N (x) = " $ % $ ( ' +1* $ k & ) % ( ' +1* & $ ) + $1 N#(+ +1)+k, 2 $ < max 1 $ $ / -, 0. + $ + $ 1. This last coditio is related to our stretch factor i Lemma 2.1. Therefore, combiig the above coditios ad simplifyig our expressio, choose " such that " < $ 2 # max 1 # # ' %, ( & # 1 + # # 1 + # ) 0 # * # - * #1 -, +1/, +1/ # k + # 1. + #. N1(0 +1)+k 17
Our choice of " esures that for all the poits i (x, x + "), ad for all 0 " k " m f k (x, x + ") # I j for some 1 " j ". That meas that the itieraries of all poits i (x, x + ") share the same iterval for at least m iteratios of f. Therefore, all the poits i (x, x + ") will have represetative sequeces i the sequece space with correspodig first m terms. Sice all the poits i (x, x + "), f k (x, x + ") # I j for some 1" j " ad all 0 " k " m, the represetative sequeces of all these poits will have first m terms as the same as the first m terms of S(x). Therefore, all poits i (x, x + ") will have represetative sequeces i the sequece space all iside the set C m = {" 0 $ 0 $ 2...}, that is, S ((x,x + ") # C m # B $ ( S(x) ). Therefore, S is quasicotiuous. The ext theorem proves the cotiuity of the iverse image of S. We have to be careful here because ot all poits i Σ have iverse images uder S i X. For example, the poit 23 ( 1) does ot have a iverse image i X. If we avoid such poits, we ca properly defie the iverse image of S ad show that the iverse image is cotiuous. We do that i the ext defiitio ad theorem. Defiitio 4.3. We defie a fuctio S "1 : # $ X by S "1 (# ) = S "1 (# 0 # 1 # 2...) = x where x " X ad x " f #i ( I $i ) for all " # $ that do ot ed i Theorem 4.4. S "1 : # $ X is cotiuous. Proof. We wat to show that "# $ %,"& > 0,'( > 0 such that S "1 B # ($) ( ) % B & (S "1 ($)). Pick " = " 0... # $. Choose " > 0. Choose M large eough such that U M = I f "i (I #i ) $ B % (S "1 (#)) (recall that we used this defiitio of U M i theorem 4.2). The, S "1 ( C m [# 0 # 1 # 2...# M "1 ]) $ B % (S "1 (#)). Now, choose " by lettig " < 1 2. The, we have show that ( m +1 S"1 B # ($)) % S "1 ( C m ) % B & (S "1 ($)) which implies S "1 : # $ X is cotiuous. M i= 0 18
Fially, we are ow i a positio to prove that S is a quasicojugacy betwee the systems ( f, X) ad (",#). Theorem 4.5. S is a quasicojugacy betwee the systems ( f,x) ad (",#). Proof. By theorems 4.1, 4.2, 4.3, 4.4 we kow that S is a quasicotiuous fuctio such that S : X " #, S "1 : # $ X is cotiuous, ad S : X " # is oe to oe ad oto (with the exceptio of a coutable umber of poits). To show that S o f = " o S, it is sufficiet to prove that "x # X, f (x) = S "1 o# o S(x). Pick x " X ad let f (x) = y. Let S(x) = " = " 0... ad the S(y) = " 3... The ( ) ( ) S "1 o# o S(x) = S "1 o# $ 0 $ 2... = S "1 $ 2 $ 3... = y. Therefore, f (x) = y = S "1 o# o S(x) ad we are doe. Sectio 5. Coclusio By showig that the systems ( f,x) ad (",#) are quasicojugate we have show f is chaotic. Without this quasicojugacy, it would have bee extremely difficult ad complicated to study the dyamical properties of f. The idea of quasicojugacy is, thus, a very useful oe. The use of quasicojugacy i our project also highlights the importace of quasicotiuous fuctios i the study of dyamical systems. The systems ( f,x) ad (",#) are ot cojugate to each other, simply because S is ot a homeomorphism. But S is a quasicotiuous fuctio with a cotiuous iverse image ad those properties ca defiitely be used to our advatage i determiig the chaotic properties of f. Future projects ca exted our work to o-liear fuctios. We could also chage the cofiguratios of the fuctio i this paper by havig differet poits of cotiuity as well as a mixture of egative ad positive slopes. It would be very straightforward to show that such fuctios are chaotic. Aother iterestig study would be to divide X ito a coutably ifiite umber of itervals. Furthermore, maps that take two dimesioal figures o the plae to other two dimesioal figures could also be studied. Fially, the ultimate goal should be to exted the results of this paper to all compact metric spaces icludig the oes metioed above. 19
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