Kirchheim, Bernd

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

Mathematica SlovacaBernd KirchheimA chaotic function with zero topological entropy having a non-perfect attractorMathematica Slovaca, Vol. 40 (1990), No. 3, 267--272Persistent URL: http://dml.cz/dmlcz/129388Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath . Slovaca 40. 1990, No. 3, 267—272 A CHAOTIC FUNCTION WITH ZERO TOPOLOGICAL ENTROPY HAVING A NON-PERFECT ATTRACTOR BERND KIRCHHEIM ABSRACT. In the paper there is constructed a continuous rnappingffrom a compact real interval to itself satisfying the following: The mapping f has zero topological entropy and there is an infinite attractor containing isolated points which are not approximate by periodic points. In the papers [4], [8] the existence of a continuous rnappingffrom the unit interval into itself with the following properties is remarked: a) The function fs of type 2X, that meansfhas no cycle of order not a power of 2 and has a cycle of any of the orders 1, 2, 22, ... b) There is a point x such that vty(x) is infinite and has isolated points. (Here vty(x) denotes the set of all limit points of the trajectory {fn(x), n > 0} generated by x. Both examples in [4], [8] are not correct. In [3] a function having the described properties a, b) and satisfying moreover the condition that all isolated points of the infinite vty(x) belong to Per (f)\Per (f) is constructed. (The existence of such an attractor is not explicitly remarked in [3] but follows from [6] where the following is shown: If ue Per(f)\Per(f), then there is some x such that ue\vf(x) and \vf(x) is infinite.) However, both examples in [4], [8] should be such that the isolated points of the infinite attractor do not belong to Per(f). In the present paper a construc-tion of such a function is given. Any function of this type is chaotic in the sense of Li and York and has zero topological entropy. Indeed, from [7] it follows thatfis chaotic if there is an infinite wf(x) and there are two distinct points u, revty(x), which are non-separable byfperiodic intervals. Letfbe a function of type 2X having an infinite \vf(x) and let u be an isolated point of viy(x). By [5] there is some point i; G Der uy(.v) (where Der M is the set of all accumulation points of M) such that no point of u'^v^ies between u and v. Then u and v have a common trajectory, see [5], and therefore are non-separable byfperiodic intervals. AMS Subject Classif icat ion (1980): Primary 58F13, Secondary 28D20. Key words: Limit points. Isolated points. Trajectory, Attractor, Entropy, Chaos. 267 1. Construction To avoid the notation of fractional numbers we will work on the interval I = [0, 69]. Our construction would also work with slightly different proportions between the used subintervals, however, these proportions cannot be chosen arbitrarily. Let / b e a continuous mapping from I into itself satisfying the conditions: l ( a ) /0 ) = 36 1(b) f{x) < 57 if x < 12 and/(x) > 12 if x < 57 1(c) f(x) = 69 - x for xe[57, 69]. The new function/is defined by the following conditions: 2(a) /(0) = 36, /(3) = 39, /(12) = 42, /(27) = 57, /(30) = 66, /(33) = 69, /(36) = 31 and/(38) = 20. 2(b) / i s linear on each [/?i5 pi + 1], i = 1, ..., 7, where /?, < ... < /?8 are the points occurring as arguments o f / in the eight statements of 2(a). 2(c) f(x) = (sofo t~])(x) for p8 < x < p9 = 61, here s or / denote the affine transformation mapping / onto [8, 31] or [p8, p9], respectively. 2(d) f(x) = 69 - x for 61 < x < 69, see the figure. 268 It is not difficult to show that/maps again /into itself and satisfies 1(a), (b), (c), clearly/is also continuous. 2. Relations between / and / Denote J = [38, 57], then clearly/([27, 31]) = [57, 67],/([57, 67]) = [2, 12] and/([2, 27]) = J. If we set a = 2, b = 31 and M = [a, b] u [57, 67], we obtain from the foregoing that for any xeM there is some n < 3 such that/"(.x)eJ. Furthermore J([b, pj) = [67, 69], /([67, 69]) = [0, 2], /([0, 2]) = [Pl, P%], and J([Pi, P&]) c M- Consequently, if x$ [p6, Pl], then there is some n < 7 such that / " ( • v ) 6 J If xeJ, then one of the following cases occurs: 3(a) f(r\x)) < 57 and then/(x)e[/>3, />4] and (r°/° r%x) = / , ( . r ) e J 3(b) /(/"'(.v)) > 57 and then/(x),/2(x),/3(x)<£f a n d ( / ° / 2 ° r ' ) ( - x ) = / 4 ( x ) e J X X This can be easily computed using s(x) = - + 8, t(x) = - +38. 3 3 Lemma. If x, /n(jc) < 57, then t(x\ t(fn(x))eJ and f2n(t(x)) = t(fn(x)). Conversely, if x, fn(x)eJ, then there exist t~l(x), t~](fn(x)) < 57 and f{n2)(t~](x)) = t~\fn(x)). In particular, Jhas a cycle of order n > 1 if and only if f has a cycle of order n/2. Proof: The first two statements immediately follow from 3(a), (b) by in-duction. To prove the third statement it suffices to show that each/— cycle of order greater than one has a point in J and that each/-cycle contains a point smaller than or equal to 57. The second assertion is by 1(c) trivial and our foregoing considerations show that if there were a n / — cycle of order n > 1 having no point in / , then this whole cycle would be contained in [F6, p7]. But this contradicts the fact that/ is on [P6, p7] a linear map with derivation smaller than — l.n 3. Existence and properties of the required function We define the mapping /0 by _ f 36 for xe[ 0, 33] ^o( = (69 - x for .ve [33, 69] Then /0 satisfies the conditions 1, moreover f\ = / J and therefore/ has cycles exactly of order one and two. Next we define by / n + l = / n (in the sense of Construction 1) a sequence of continuous functions satisfying the conditions 1. 269 It follows by induction that for n > 1, m > n and any x£tn~ '([P8, 57]) we have jn(.v)=/m(.v). Hence, we conclude 1 1 / - / J | x < A(8n~'([0, 69])) = 69/3n~ ', where A denotes the Lebesgue measure. Now we p u t / = lim / . Since t h e / n -»• x converge uniformly t o / / i s a continuous map from I into itself and (by [1]) of type V-. Indeed, for each n > 1 tn(pF) belongs to a n / — cycle of order 2n, here pFdenotes the (common) unique fixed point of the /^s , n > 1, which is contained in [/?6, p7]. On the other hand any m a p / has no cycle of order greater than 2n. Clearly / satisfies also the conditions 1. Because / = / , from the foregoing Lemma the following important properties of the m a p / f o l l o w : 4(a) If .\\/n(.v) < 57, then /(.v), t(fn(x)) < 57 and / 2 n ( / (x ) ) = / ( / n (x ) ) . 4(b) If . \ \ /n( .v)G7, then r](fn(x)) =fn2(r](x)) 5. If .w v e / . then y =/( .v) ifl\v( v) = /(r(.v)). Theorem 1. There is a continuous mapping ffrom the interval I into itself of type 2V and a point xe I such that the at tractor \\)(x) is an infinite set having isolated points. These isolated points do not belong to P e r ( / ) , where Per(f) denotes the set of all periodic points of the function f Proof . We c h o o s e / t o be the function constructed above and x to be the point b = 31. Further denote c = /?:, d = p5, an = tn(a) and similarly for bn, cn, dn. Because/(.v) =f(\) = d + (x — d,)forxe[d,, l(/?6)], we obtain by induction from 4(a) the following: 6. fn](x) = dn_ , + (A- - dn) for n > 1 and dn < x < /n(F6). Similarly from /(.v) = / (A) = C] + (x - c) for P] < x < c. 7. fn(x) = cn _ , + (A - cn) for n > 0 and /"(/?,) < A < cn follows. Because f(b) =f(a) =f(t(p})) =- t(f(px)) = t(b) = />„ from the property 4(a) we get bys induction that 8. bn_, =/"-n(/3n) for n > 0. Denote S= [J t"([P}. p,)). S= [J /k([^„ p2]u[p„ p4]u[d, b]vj{Pl}) u [57, 69] and F = [/?,. c] u [P-] u [d,, t(p6)]. We obtain 9. \J f(tn(T))cz 5 for /?> 1. k = i Indeed, in case n = 1 the statement 9. is evident and now assume that it holds for some n > 1. For each point xetn~](T) there is some yetn(T) satisfying A = /( r). If 1 < k < 2n'\ then either k = 2m for some integer m and in this case /k(.v) = / : m ( / ( ; • ) ) - t(fm(y))et(S) c S by assumption, or k = 2.m + 1 and then /k(.v) = / ( / : m ( . v ) ) e / ( / ( S ) u /n ~ ](T)) c: [/7^ ^ ] ; remark that n + 1 > 2. Furthermore, from f(F-) = b a n d / ( P , ) = p7 it follows by 4(a) that 10. fn(tn(P])) = tn(p+fn(tn(Pl)) = tn(b) = bn if n > 0. Finally, the statements 6.. 7., 9.. and 10. imply 270 11. f(b) c- S, hence f\b)$ Q ^((c, Pi) u (p4, d)) for k > 0. i = 0 Note thatf2n+ \bn) = an a n d f2 » = tn+\P]). Iterating the properties 6., 8. and using the fact that lim (bn — dn) = 0 we get n -* x immediately {dn, n > 0} c= ny(b). Hence vvy(b) contains also the pointsf(d) = 66, f2(d) = c, and by statement 7. also all cn, n > 1. Now take some n > 1, let fk(b)e [tn(p}), cn] and let / be the largest integer not greater than k with f(b) = bm, m > 0. Then the statements 7., 9., and the equalityf2(x) = c — (x — d) if d<x<b imply cn-f(b) = bm - dm = (b - d)/3m. For / = min{ j>k , f(b)e[tn(P]), cn]} we get similarly cn -f(b) = (cn -f(b))/3. By statement 11. cn is therefore an isolated point of wf(b). Now denote Orb = {dn}n = 0 u {66} u u {cn}*= o, remark that according to 6. and 7. all these points belong to one orbit off Since in the foregoing consideration n was chosen arbitrarily and since f(vty(b)) = viy(b), it follows that each point of Orb is an isolated point of uy(b). To finish the proof we show at first that Per(f) cz [/?3, p4] u {pF} u [42, 57]. Assume that k is the smallest positive integer such that there is anf-cycle x,, ..., x>k with .Y, e (p4, p6). From the proof of the Lemma we already know that there exists the largest / not greater than 2k such that xteJ = [38, 57]. Now the statements 3a), b) imply that f(l_1(xi)) > ~l,f(x) = .v, and that /"'(.Y,), f(t~\x)), and t~\x}) belong to anfcycle of order 2k~ l which is not contained in [p}, p4]uJ again. But this contradicts the choice of k and implies that Per(f) c: [p3, p4] u {pF} u / . The properties 4. yield now [42, 57] => l(Per (f)) = = Per(f) n J. This means Per(f) c Per(f) cz [p3, p4] u {pF} u [42, S7]. Fur-thermore we obtain /(Per(f)) = Per(f) n [42, 57] and since in general f(Per(f)) = Per(f), from C,^Per(f) we directly get the required statement Orb n Per(f) = 0. Theorem 2. For the at tractor wf(b) considered in the proof of Theorem 1 more precisely, the following holds. Let wf(b) = Pu D be the Cantor—Bendixson decomposition ofwf(b), then the perfect kernel P is an affine image of the familiar "middle third Cantor set" satisfying min P = 12 and max P = 57. Moreover D is the set of all isolated points ofwf(b) and forms a "two-directionaP trajectory off that means f(D) = D and for any x, y e D there is some m > 0 such that f~ (x) = y orfm(y) = x. P r o o f First we show that t(wf(b) n [0, 57]) = wf(b) n [38, 57] and s(wf(b)n[\2, 57]) = uy(A)n[12- 27]. Hence, let xewf(b)n[0, 57]. If .v = 57, then t(x) = x and in case of x < 57 the conclusion t(x)ewf(b) follows im-mediately from 4a) and the fact that t(b) = b] =f(b). Conversely, if i'env(b)n[38, 57], then 4a), and 11. give / ](x)ewf(b). Next, if ve[12, 57]n n wf(b), then clearly some xewf(b)nf~\y) exists, hence x < 57. From 5. we now obtain s(y) =f(t(x))ewf(b), as shown before. Finally, assume ye w((b) n 271 n [12, 27]. Thenf(y) = y + 30G 117(b) n t38^ 571 a n d consequently f(y) = t(x) for some xe 117(6)0 [0, 57]. This implies s(x) = t(x) — 30 = y and proves the last inclusion. The proof of Theorem 1 moreover shows that 117(b)n IP- ' 2 ) = M> 117(b) n (27^ 33] = {d}, 117(b) ^ t38> 42) = fa}, and 117 (b) n (57, 69] = {66}. Now according to 11. Pc=[12, 27]u[42, 57] and P = s(P)ut(P). Under these assumptions the theory of selfsimilar sets, see [2], yields that P must be an affine image of the standard Cantor set with position as described in the assertion. Note that 8, t satisfy the "open set condition". Recall that P was chosen to.be the set of all condensation points of 117(b), but it is easy to see that also for the set P of all nonisolated points of 117(b) P = S(P)KJ t(P) holds, hence we get p = pm We finish the proof by showing that D = D with D = {f(dk _,,); k > 0, 0 < i < 2k} u {d} u {66} u {/(ck); k > 0, 0 < / < 2k}. Using the properties 4., 5. one can conclude that t(D n [12, 57]) cz D and s(D n [12, 57]) cz I5. Clearly each inner component interval of I P is an image of (27, 42) under some finite composition of the mappings s and t. Since (27, 42) contains two points of L>, each inner component interval /does too and therefore, by Theorem 1 in [5] / n D = / n Z). The required identity D = D now follows from the simple re-mark that D [12. 57] = D [12, 57] = {3, 66}. Acknowledgment. I want to thank J. Smital for bringing this problem to my attention and for his very helpful comments. REFERENCES [1] BLOCK. L.: Stability of periodic orbits in the theorem ofSarkovskii. Proc. Amer. Math. Soc. 82. 1981. 333—336. 2] FALCONER. K. J.: Geometry of Fractal Sets. 1st ed. Cambridge University Press 1984. 3] H S I N C H U - X I O N G J I N G C H E N G : A counterexample in dyaynamical systems of [0, 1]. Proc. Amer. Math. Soc. 97. 1986. 361—366. 4] KENZEGULOV. CH. K — SARKOVSKII. A. N.: On properties of the set of limit points of an iterated sequence of continuous functions (Russian). Volzsk. Mat. Sb. 3, 1965, 343 -348. 5] SARKOVSKII. A. N.: Attracting sets containing no cycles (Russian). Ukrain. Mat. Zurn. 20, 1968. 136—142. 6] SARKOVSKII. A. N.: On a theorem ofG. D. BirkhofT(Russian). Dopov. Akad. Nauk USSR, 1967. No. 5. 429—432. 7] SMITAL. J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc. 297, 1986. 269—282. 8] VEREJKINA. M. B.—SARKOVSKII. A. N.: Recurrence in one-dimensional dynamical sys-tems, in Approx. and Qualitative Methods ofthe Theory ofDiflerential & Functional Equations (Russian). Instil. Math. AN USSR. Kiev 1983. pp. 35—46. Received Jul\ 30. 1987 Cstav aplikovanej matematiky a vypoctovej teclmiky UK Mlynska dolina H42 15 Bratislava 272 

A chaotic function with zero topological entropy having a non-perfect attractor

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A chaotic function with zero topological entropy having a non-perfect attractor

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