Kuchta, Milan

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Characterization of chaos for continuous maps of the circle

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae UniversitatisCarolinaeMilan KuchtaCharacterization of chaos for continuous maps of the circleCommentationes Mathematicae Universitatis Carolinae, Vol. 31 (1990), No. 2,383--390Persistent URL: http://dml.cz/dmlcz/106868Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copyof any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronicdelivery and stamped with digital signature within theproject DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czComment .Math .Univ.Carolinae 31,2(1990)383-390 3 8 3 Characterization of chaos for continuous maps of the circle MILAN KUCHTA Abstract. Recently Li & Yorke introduced the notion of chaos for continuous maps . The recent papers by Jankova, Smi't.al and Kuchta, Smital give a characterization of chaos for continuous maps of compact real interval into itself. In the present paper we give a characterization of chaos for continuous maps of the circle. The class of continuous maps of the circle extends the class of continuous maps of the interval, if we consider dynamica l systems generated by the maps from these classes. Keywords: Chaos, circle, iteration, topological dynamics Classification: 54H20, 26A18 1 . INTRODUCTION Let M denote a compact real interval or a circle and C(M, M) denote the set of continuous maps of M into itself. Let / € C(M, M); 8 > 0 and A C M be a set such that for any ar, y £ A; x ^ y and any periodic point p of / : (1) lim8up||r(x),/"(v)||>* n—+oo (2) liminf||r(z),r(y)||=0 n—*oo (3) l i m s u p | | / n ( z ) , / n ( p ) | | > * n—+-oo Here fn is the n-th iterate of / and ||x, y|| is the distance of points xy y. Then A is called a scrambled set of / , or (when 8 > 0) a ^-scrambled set. The function / is chaotic in the sense of Li & Yorke [7] if / has an uncountable scrambled set. But, indeed, there are some other definitions of chaos. This paper gives a characterization of chaos for the maps from C(M, M). Before we state the results, we recall some terminology. Let / G C(M,M). We say that an interval J C M is an /-periodic interval of period k € N if / * ( / ) = J and /'(J) PI f>(J) = 0 for t ^ j ; ij = 1 . . .*. Interval J may be degenerated to a point and be called a periodic point. Let Per(f) denote the set of all periodic points of / and P(f) denote the set of all periods of periodic points of / . Two points u>v € M are /-separable if there are disjoint periodic closed intervals Ju,JvCM (may be degenerated to a point) with u € Ju; v € Jv. Otherwise u,v are /-nonseparable. The set of all limit points of a trajectory {f(x)}tLi -s called the attractor of / and x, and is denoted by u(f,x). Now we give main results of this paper. 384 M.Kuchta Theorem A. Let f € C(M,M) and Per(f) ^ 0. The following conditions are equivalent: (a) / is chaotic in the sense of Li & Yorke; (b) / has an infinite attractor containing two f-nonseparable points; (c) for some 6 > 0, f has a nonempty perfect 6-scrambled set; (d) / has a trajectory which is not approximated by cycles; (e) / is topologically conjugated to a function, which has a scrambled set of positive Lebesgue measure; ({) for some 6 > 0, f has a nonempty 6-scrambled set; (g) / has a scrambled set containing two points. Theorem B . Let f € C(M,M) and Per(f) = 0. Then f is topologically semi-conjugated to a rotation of a circle and f has no two-point scrambled set. Remark 1.1. Theorem A was proved in papers by Jankova, Smftal [5] and Kuchta, Smftal [6] for maps of compact real interval into itself. 2.MAPS WITH PERIODIC POINTS Let N, Z, R denote the sets of natural, integer, real numbers respectively and 5 = R\Z denote the circle. We shall denote by n : R —> 5 the natural projection. Let xi,x2 € 5. We define: ||ori9a;2|| = min{|yi - 2/2I; £i = n(y,)} Let F € C(R, R) and / € (7(5,5). Then F is a lifting of / if / o n = n o F. Let F be a lifting of / . Then there is kf € Z such that for any x € R,z € Z: F(x + z) = F(x) + zkf. So we can define the degree of / : deg(f) = kf Let / € C(S, S), deg(f) = 1 and F be a lifting of /.Then: p(F, x) = limsup ~(Fn(x) - x) for x € R n—*oo n p(F)={p(F,x);x€R}. The class C(S, S) can be decomposed into following disjoint sets (see [1], [2], [8]): Wo = {/€C(5,5);Per(/) = 0} Wi = {/ € C(S, S); P(fn) = {1} for some n € N} W2 = {/ € C(S,S)',P(fn) = {2*;t = 0,1,2,...} for some n € N} Wz = {/ 6 C(S,S)',P(fn) = N for some n € N} Remark 2.1. Let / € C(S,S) and n € N. It is easy to see that / has some property from the properties (a) , . . . ,(g) from Theorem A if and only if fn has the same property. Lemma 2.2. Let f € C(S,S) and 1,3 € P(f). The following conditions hold: (i) for some 6 > 0, / has a nonempty perfect 6-scrambled set; (ii) / has an infinite attractor containing two f-nonseparable points. PROOF : Without any loss of generality we may assume that: 0 < xj < x2 < x3 < 1 and /(II(0)) = 11(0); /(n(x,)) = H(x2); /(IT(x2)) = n(xs); Characterization of chaos for continuous maps of the circle 385 / (n (x 3 ) ) = II(xi). ( {n(x i ) ,n(x 2 ) ,n(x 3 ) } is three cycle and n(0) = 11(1) is fixed point.) It is easy to see, that for any given function / there exists a code, consisting from four numbers (0 or 1), such that Table 1 gives us an information about properties of / . Then according to Table 2 we obtain from this code some intervals h>h and some m € N, such that for all n € N and s» € {0; 1} there exists a closed interval Iai...9n±* such that: *n + l -".,.....+, C /.,....„ and / m (П(/ . l . . . .„ + 1 ) ) = П(/...... n + 1) Let I, = П h for every s = {si}^Lx $i € {0; 1}. Let A = {Ia; I9 consist of one point}. There exists such B C A that U(B) is nonempty perfect 8-scrambled set (6 > 0) for function fm (see [5]). There exists x € S such that n(A) C u>(/m,x) (for more details see [3], Symbolic dynamic, pp.39-43). For any y € 11(1?) and J C S closed k /m-periodic interval of period k € N it holds: if y € J then w(fm,x) C U /m i(J) (y is not periodic). Because for any two points of 11(B) (2) holds for function / m we obtain 11(B) C J and this proves that any two points of 11(B) are /m-nonseparable. Now it suffices to consider Remark 2.1 and we are done. • Table 1 Code: Property: * = There exists an interval 0 . . . * Ao C [0;xi]; /(П(Л0)) = П([0;x2]) 1 . . . * Ai C [0;xi]; /(П(Лi)) = П([x2;l]) . 0 . . * B0 C Џx;x2]; /(П(Ј?o)) = П([x2;x3]) . 1 . . * Bi C[xt;x2]; /(П(Бi)) = П([xз;l + x2]) . . 0 . * Co C [x2;x3]; /(П(Co)) = П([x3;l + xi]) . . 1. * CiC [x2;x3]; /(П(Ci)) = П([xi;x3]) . . . 0 * DoC [x3; l]î /(П(.D0)) = П([xi;l]) . . . 1 *£>iC[x3;l]; /(П(Di)) = П([0;xi]) Table 2 Code: Eesult: 0 1 . . m = 1, /o = [0;xj], h = [xi;x2], . 0 1 . m = 2, Һ = [*i;я2], Һ =[*2ÎЯзL 0 0 0 . m = 3, Io = [0;xi], Һ = [2lî22]» . . 00 m = 2, Һ =[Я2Î*3]» Ii = [x3;i], 1 . 0 1 m = 2, h = þгî^зjł li = [x3;i], 1 1 1 . m = 3, h = [21; £2], Һ = [22; 23], 386 M.Kuchta Lemma 2.3. Let f € C(S, 5) , / € Wx U W2 and f have a fixed point. Then there exists F lifting of f and a closed bounded interval I C R greater than one, such that F\ieC(i,i). PROOF : We know that |deg(/) | < 1 (see [2]), so we can divide our problem to the three cases. * Let deg( f) = 0. In this case F is periodic and we are easily done. * Let deg( f) = 1. There is F lifting of / such that F has a fixed point. It follows that 0 € p(F). Since / € W\ U W2 the set p(F) must have only one point (see [4]). Then p(F) = {0}. We claim there exists a point a € R such that: if x > a then F(x) > a. If it is not true, then there exists S > 0 such that for every x G R there is some y > x -f S such that F(y) = x. Now we can easily obtain that there is z € R such that p(F, z) < —S which is impossible. In a similar way we can find b € R such that: if x < b then F(x) < b. Since the properties of a,b are not changed by addition of an integer we can now define J = [a; b) where b — a > 1. * Let deg(f) = —1. Let F be a lifting of / . We know that F has a fixed point, deg(f2) = 1 and F2 is the lifting of f2. Let J be an interval for function F2 which contains some fixed point of F and F2(J) C J as we found it in the previous case. Now we can define I = J U F(J). m Lemma 2.4. Let f € C(S, 5), F be a lifting of f and I C R be the closed bounded interval greater than one such that F\ I € C(I,I). Then the following conditions hold: (i) / has an infinite attractor containing two f-nonseparable points if and only if F\I has an infinite attractor containing two F \ I-nonseparable points; (ii) if f has a nonempty S-scrambled set then F\I has a nonempty S-scrambled set; (iii) if f has two-point scrambled set then F\I has two-point scrambled set; (iv) if for some SF > 0 function F\I has a nonempty perfect Sp-scrambled set then there is Sf > 0 such that f has a nonempty perfect Sf-scrambled set. PROOF : It is easy to see: (4) u(f, n(x)) = Il(u>(F, x)) for all x 6 R (5) P e r ( / ) = n ( P e r ( F \ I ) ) (6) P(f) = P(F\I) (7) \deg(f)\ < 1 (i) Let us check the following possibilities: Characterization of chaos for continuous maps of the circle 387 Let f €Wn. This is impossible (see (6)). Let f G W*. Then u>(F, x) is finite for all x € R (see [11]) and also u(f, y) is finite for all y € S (see (4)). Let / € W%. In this case functions f,F\I have infinite attractors containing two nonseparable points (see Remark 2.1, Lemma 2.2 and [5]). Thus the only interesting case is f £ Wo. Let x € I and a, b € u>(F, x), a£b. If a, b are F \ /-separable then either a, b are periodic points and (see (5)) n(a), 11(b) are also periodic and they are /-separable (if n(a) = U(b) then they are not two /-nonseparable points), or there exist a closed periodic interval J C I and n € N such that a € J, b € Fn(J) and J n Fn(J) = 0. But n(J) is also periodic and U(J) 0 / n ( n ( J ) ) = 0 and so we have that 11(a), U(b) are /-separable. If a, b are F-nonseparable then there is an interval J C I such that a, b € J and F*(J) fl F>(J) = 0 for all » ^ j (see [10]). Then n(a), n(6) G U(J) and /'(II(J)) 0 />(J) = 0 for all i ^ j . It follows that n(a) ?- n(6) and they are /-nonseparable. Now it suffices to consider (4) to finish the proof of (i). (ii) Let x € I and {II(x)} be a 8-scrambled set for / . Then (5) implies that {x} is a 8-scrambled set for F\I. (iii) Let x,y e I and {U(x), II(y)} be a scrambled set for function / . Then there is an increasing sequence {n,}?^, m 6 N such that lim \Fni(x) —Fni(y)\ = k »—->oo for some k € Z (see (2)). Let us discuss the three cases (see (7)). Let deom = 0. Then lim |Fn*+1(ar) - Fn<+%(y)\ = 0 and now it is easy t—>oo to see that {x; y} is the scrambled set for F\I. Let deg(f) = 1. Then because I is greater than one there is m 6 N and &i ,k2 € Z such that: F^x + k^F^y + k^el and lim |Fn<(* + h) - Fni(y + * 2 ) | = 0. t—*-oo Then {Fm(x -f *i), Fm(y + ^2)} is the scrambled set for function F\I. Let dea(f) = - 1 . Then deg(f2) = 1 and {U(x), IL(y)} is the scrambled set for f2 and F 2 is the lifting of f2 and we are in the previous case. Now it suffices to consider Remark 2.1 to finish the proof of (iii). (iv) Let D C I be a nonempty perfect ^-scrambled set for function F\I (Sp > 0). Because F \ / is continuous in the compact interval I we can show that: * there exists S\ > 0 such that for any x, y € D, x ^ y there is a limit point of the sequence {||Fn(x),Fn(|/)||}SL1 in the open interval (Si; 1 - £1); * there exists 62 > 0 such that for any x € D, p € Per(F \ I) there is a limit point of {\\Fn(x), Fn(p)||}^L1 in the open interval (k + S2; k + 1 - S2) for some k € Z. 388 M.Kuchta Now we can take 8/ = min{8i; £2} and n (D ) is the nonempty perfect 8/-scrambled set for / . • PROOF of Theorem A: We can assume that / 6 C(S, S) (see Remark 1.1). If / € W\ U W2 then Theorem A follows from Remarks 1.1, 2.1 and Lemmas 2.3, 2.4. If / € W3 then Theorem A follows from Remark 2.1 and Lemma 2.2. • Corol lary 2.5 . The set W% contains only functions which have only trajectories which are convergent to periodic orbits. The set W2 can be decomposed into two disjoint subsets: Chaotic functions and functions that have only trajectories approximated by cycles. The set W3 contains only chaotic functions. 3 .MAPS WITHOUT PERIODIC POINTS Let / € C(S, S), deg(f) = 1 and F be a lifting of / . We shall call a set A C S a m a t set of/ (minimal almost twist) if A is non-empty, closed, invariant (f(A) = A), minimal (for every a: € A; v(f,x) = A), almost twist (F\U"1(A) is nondecreasing) and there is p(A) € R such that p(F,y) = p(A) for every y 6 n~"1(A). Remark 3.1. It is easy to see that if / 6 C(M,M) and Per(f) = 0 then / € C ( S , S ) and deg(/) = l. L e m m a 3.2. (Misiurewicz [9]). Let f € C(S,S), deg(f) = 1 and F be a lifting of f. Then for every a € p(F) there exists A C S a mat set of f such that p(A) = a. Lemma 3.3. (Misiurewicz [9]). Let f € C(S,S), deg(f) = 1 and A be a mat set off. Then: if p(A) is rational then A is a finite set; if p(A) is irrational then: (i) either A = S or A is homeomorphic to the Cantor set; (») if x € A is an endpoint of an interval disjoint from A, then there exists a unique y € A with f(y) = x; this y is also an endpoint of an interval disjoint from A; (iii) if x € A, then either there is a unique y € A with f(y) = x} or there are two such points; in this case they are the endpoints of some interval disjoint from A. Corollary 3.4. Let f € C(S,S), deg(f) = 1, F be a lifting of f and A C S be a mat set of f. Then for all x,y € IT"1 (A) it holds: (i) if (*, y) n n-H-A) = 0 then (F(x), F(y)) C\ Il^A) == 0 (ii)[F(x),F(y)]cF([x,y]) (iii) if A~ S then F is increasing Lemma 3.5. Let f € C(S> S), deg(f) = 1, F be a lifting of f and Ac S be a mat set of f homeomorphic to the Cantor set Let, for all x,y € II~"1(A), hold: if (x, y) fl W\A) = 0, then F([x, y}) = [F(x), F(y)}. Characterization of chaos for continuous maps of the circle 389 Then there is a semi-homeomorphism h £ C(S, S) such that: (i) h(x) = h(y) if and only if there is an interval J C S such that int(J)f\A = 0 and x,y £ J; (ii) h o f = g o h where g £ C(S, S) such that deg(g) = 1 and S is the mat set of function g. PROOF : Function h is a well known function of Cantor's type such that h(A) = S and deg(h) = 1. • Lemma 3.6. Let f £ C(S,S), deg(f) = 1 and S be a mat set of f. Then f is topologically conjugated to the irrational rotation of circle. PROOF : Let F be a lifting of / . Then p(F) = {a} where a is irrational (see Lemma 3.3) and there is x £ R such that F(x) = x -f a. We can assume that F(k) = k F a for all k £ Z. Now we define: H(Fn(k)) = na-r k for all n, k £ Z (F is homeomorphism). Let nx,n2,z,r £Z and Fni(z) > Fn*(r) then (deg(f) = 1) Fn--n*(0) > r - z then (F is increasing) jF*(»i-»a)(o) > fc(r _ 2 ) then (k goes to infinity) />(ir,0)(n1 — n2) > r — z then (p(F, 0) = a is irrational) ani + z > an2 F r. So we have that function H is increasing on the set D C R which is dense in R (S is the mat set) and H(D) is dense in R (a is irrational). Function H can be extended to a function from C(R, R) and so we have H o F = G o H where G(x) = x + a for x £ R. Here H, G are the liftings of fc, gr respectively, where & is a homeomorphism, g is irrational rotation of circle and ho f =: goh. • Lemma 3.7. Let f £ Wo, F be a lifting of f and A C S be a mat set of f. Then for all x,y£ W^A) it holds: If ( x , y ) n r 1 ( A ) = 0 then F([x,y]) = [F(x),F(y)} PROOF : Let /* £ C(S, S) be a function such that if F* is a lifting of /» then for all a;,y € II"~1(.A) it holds: (i) F.(x) = F(x); (ii) if (*, y) n li^(A) = 0 then Fm([x, y]) = [F.(x), F«(y)}. Since p(A) is irrational ( / £ W0), from Lemmas 3.3, 3.5, 3.6 we obtain that for every closed intervals J, K C S such that, mt(J)nA = 0 and mt(K)nA^0 there is such n £ N that J C f?(K) (A is the mat set of /* ) . If there is a closed interval J C S such that: m<(J)nA = 0 and mt(/( J)) n A ̂  0, 390 M.Kuchta then there is n € N such that J C / n ( / ( J ) ) and from here we obtain that J C / n + 1 ( « 0 which is impossible because Per(f) = 0. So we have int(f(J)) H A = 0. • PROOF of Theorem B: Prom Lemmas 3.2, 3.5, 3.6, 3.7 it follows that / is topolog-ical^ semi-conjugated to the irrational rotation of the circle. Let A be a mat set of / . The semihomeorphism h has the properties from Lemma 3.5 and h o / = g o h where g is the rotation of the circle. Now we assume that liminf | | / n (x ) , / n (y ) | | = 0 for some x,y € S. f l—*oo So we have liminf ||/.(/n(x)),ft(/n(y))ll = «. Uminf || f f"(fc(i)), f f"(%))|| = 0 n—*oo n—->oo and because g is the rotation of the circle we have h(x) = h(y). FVom here we obtain that there is an interval J C S such that x,y £ J and int(J) f) A = 0. FVom Lemma 3.7 we obtain that int(Jn) 0-4 = 0 (Jn = fn(J)) for every n € N. Because fn(J) f) fm(J) = 0 for n ^ m (Per(f) = 0) we have: l imsup | | / n (x ) , / n (y ) | |=0 . n—>oo Now we are completely done. • REFERENCES [1] L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc. 8 2 (1981), no.3, pp . 481-486 . [2] L. Block, J. Guckenheimer, M. Misiurewicz, L. S. Young, Periodic points and topological en-tropy of one~dimensional maps, in book: Global theory of dynamical systems, (Proc. Inter-n a l Conf., Northwestern Univ., Evans ton, 111., 1979, p . 18-34. Lecture Notes in Math. 812, Springer, Berlin 1980 [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second Edition, Addison-Wesley, New York 1989. [4] R. Ito, Rotation sets are closed, Math. Proc. Cambridge Philos. Soc. 89 (1981), no . l , pp . 107-111. [5] K. Jankova, J . Smital, Characterization of chaos, Bull. Austral. Math. Soc. 3 4 (1986), no.2, pp . 283-292 . [6] M. Kuchta, J. Smital, Two point scrambled set implies chaos, in book: European Conference on Iteration Theory, (ECIT 87), World Sci. Publishing Co., Singapore. [7] T . Y. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no.10, pp . 985-992. [8] M. Misiurewicz, Periodic points of maps of degree one of a circle, Ergod. T h . & Dynam. Sys. 2 (1982), no.2, pp . 221-227. [9] M. Misiurewicz, Twist sets for maps of the circle, Ergod. Th. & Dynam. Sys. 4 (1984), no.3, pp . 391-404. [10] J. Smital, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. l , p p . 269-282. [11] A. N . Sarkovskif, On cycles and the structure of a continuous mapping, Ukrain. Mat. Z. 17 (in Russian) (1965), pp . 104-111. Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, 814 73 Bratislava, Czechoslovakia (Received March 8,1990) 

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Characterization of chaos for continuous maps of the circle

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