Nonmonotonicity of kneading invariants in the family of kinked maps

Nonmonotonicity of kneading invariants in the family of kinked maps

We study monotonicity properties of the kneading invariant for one-parameter families of piecewise linear unimodal maps, and prove a theorem on violation of monotonicity of the kneading invariant for maps that are symmetric, convex, and consist of four linear pieces. The fact that such maps can not...

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Datum:2005
1. Verfasser: Volkova, O.Yu.
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Sprache:English
Veröffentlicht: Інститут математики НАН України 2005
Schriftenreihe:Нелінійні коливання
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Zitieren:Nonmonotonicity of kneading invariants in the family of kinked maps / O.Yu. Volkova // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 159-164. — Бібліогр.: 14 назв. — англ.

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spelling irk-123456789-1778832021-02-19T01:26:35Z Nonmonotonicity of kneading invariants in the family of kinked maps Volkova, O.Yu. We study monotonicity properties of the kneading invariant for one-parameter families of piecewise linear unimodal maps, and prove a theorem on violation of monotonicity of the kneading invariant for maps that are symmetric, convex, and consist of four linear pieces. The fact that such maps can not be approximated with smooth mappings that have negative Schwarzian is proved using a dynamics argument. Дослiджуються властивостi монотонностi нiдинг iнварiанту для однопараметричних сiмей кусково-лiнiйних унiмодальних вiдображень. Доведено теорему про порушення монотонностi нiдинг iнварiанту для вiдображень, якi симетричнi, опуклi та складаються з чотирьох лiнiйних кускiв. Наведено динамiчне доведення факту, що такi вiдображення не можна апроксимувати гладкими вiдображеннями з вiд’ємним шварцiаном. 2005 Article Nonmonotonicity of kneading invariants in the family of kinked maps / O.Yu. Volkova // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 159-164. — Бібліогр.: 14 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177883 517.9 en Нелінійні коливання Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We study monotonicity properties of the kneading invariant for one-parameter families of piecewise linear unimodal maps, and prove a theorem on violation of monotonicity of the kneading invariant for maps that are symmetric, convex, and consist of four linear pieces. The fact that such maps can not be approximated with smooth mappings that have negative Schwarzian is proved using a dynamics argument.
format Article
author Volkova, O.Yu.
spellingShingle Volkova, O.Yu.
Nonmonotonicity of kneading invariants in the family of kinked maps
Нелінійні коливання
author_facet Volkova, O.Yu.
author_sort Volkova, O.Yu.
title Nonmonotonicity of kneading invariants in the family of kinked maps
title_short Nonmonotonicity of kneading invariants in the family of kinked maps
title_full Nonmonotonicity of kneading invariants in the family of kinked maps
title_fullStr Nonmonotonicity of kneading invariants in the family of kinked maps
title_full_unstemmed Nonmonotonicity of kneading invariants in the family of kinked maps
title_sort nonmonotonicity of kneading invariants in the family of kinked maps
publisher Інститут математики НАН України
publishDate 2005
url http://dspace.nbuv.gov.ua/handle/123456789/177883
citation_txt Nonmonotonicity of kneading invariants in the family of kinked maps / O.Yu. Volkova // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 159-164. — Бібліогр.: 14 назв. — англ.
series Нелінійні коливання
work_keys_str_mv AT volkovaoyu nonmonotonicityofkneadinginvariantsinthefamilyofkinkedmaps
first_indexed 2025-07-15T16:06:54Z
last_indexed 2025-07-15T16:06:54Z
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fulltext UDC 517 . 9 NONMONOTONICITY OF KNEADING INVARIANTS IN THE FAMILY OF KINKED MAPS* НЕМОНОТОННIСТЬ НIДIНГ IНВАРIАНТIВ ДЛЯ СIМ’Ї КУСКОВО-ЛIНIЙНИХ ВIДОБРАЖЕНЬ O. Yu. Volkova Inst. Math. Nat. Acad. Sci. Ukraine Tereshchenkivs’ka Str., 3, Kyiv, 01601, Ukraine e-mail: oxana@imath.kiev.ua We study monotonicity properties of the kneading invariant for one-parameter families of piecewise linear unimodal maps, and prove a theorem on violation of monotonicity of the kneading invariant for maps that are symmetric, convex, and consist of four linear pieces. The fact that such maps can not be approximated with smooth mappings that have negative Schwarzian is proved using a dynamics argument. Дослiджуються властивостi монотонностi нiдинг iнварiанту для однопараметричних сiмей кусково-лiнiйних унiмодальних вiдображень. Доведено теорему про порушення монотонностi нiдинг iнварiанту для вiдображень, якi симетричнi, опуклi та складаються з чотирьох лiнiйних кускiв. Наведено динамiчне доведення факту, що такi вiдображення не можна апроксимувати гладкими вiдображеннями з вiд’ємним шварцiаном. 1. Introduction. Let fa : [0, 1] → [0, 1], a ∈ [a1, a2], be a family of maps such that fa = a · f, where f is a unimodal map with a unique critical point c, i.e., a continuous map that is strictly increasing on the left of c and strictly decreasing on the right of c. We assume that f(0) = = f(1) = 0. The kneading invariant of fa is an infinite sequence K = e1e2e3 . . . of symbols 0, C, and 1 defined by ek =  1 if fk a (c) > c, C if fk a (c) = c, 0 if fk a (c) < c. For sequences of symbols 0, C, 1, there is a signed lexicographical order ≺ defined in the follo- wing way: if K and K ′ coincide up to entry n − 1 and en ≺ e ′ n (where 0 ≺ C ≺ 1), then K ≺ K ′ (K � K ′ ) if the number of the symbol 1 in e1 . . . en−1 is even (odd). We are interested in the problem: Is K(fa) a monotone function of a? (1) The same question is asked for topological entropy htop of fa, i.e.: is htop(fa) a monotone function of a? It is well-known that the topological entropy of a unimodal map depends only on its knea- ding invariant and the dependence is monotone with respect to the signed lexicographical order ∗ The research was supported by a visitors grant of the London Mathematical Society. c© O. Yu. Volkova, 2005 ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 159 160 O. YU. VOLKOVA indicated above (see [1]). Hence, monotonicity of kneading invariant implies monotonicity of topological entropy. The problem (1) concerns the phenomenon that unimodal maps become more complicated topologically if the map increases. This phenomenon was observed in many systems in physics and biology (see e.g. [2, 3]). But in general (1) is not true. A positive answer is known only for few families of unimodal maps. In particular, nondecreasing of kneading invariant was proved for several polynomial families, fa(x) = ax(1 − x) [1, 4, 5], fa(x) = a − xl, for l = 4, 6, 8, . . . [6] and fa(x) = ax2(1 − x) [4]. Proofs for these families use methods of complex analysis and no "real"proof is known. Note that the maps from the families indicated above have negative Schwarzian derivative. A positive answer to the problem (1) is known for several families of piecewise linear maps [4, 5, 7]. In particular, for the symmetric family of piecewise linear maps with 2 pieces (so called tent maps) the kneading invariant is strictly increasing with respect to the parameter. Note that the piecewise linear maps have zero Schwarzian derivative. However, several counter examples have been found. The first example was given by Zdunik [8]. She constructed piecewise-linear maps which consist of 8 linear pieces, convex, symmetric and cannot be approximated in C0 topology by maps with negative Schwarzian derivative. Other examples are due to Nusse and Yorke [9] and Kolyada [10], but the map which they constructed is not convex. Other examples can be found in [11]. Due to these examples, for the family of maps fa = a · f the standing conjecture is: Conjecture 1. If fa = a · f is a family of convex unimodal maps with negative Schwarzian derivative then K(fa) is a nondecreasing function of a. In this paper we improve on Zdunik’s result in the sense that we show that the kneading invariant can be nonmonotone for unimodal maps which are convex, symmetric, but consist of 4 linear pieces. We call such maps one-kinked maps (for precise definition see Section 2). We should point out that nonmonotonicity of kneading invariant among (one)-kinked maps was known. For example, in [12] (Chapter 9) it was mentioned that Bielefeld found a counter example to monotonicity of the kneading invariant for some families of kinked maps. However, to our knowledge, no explanation why this is true was ever published. In this paper we show the mechanism why nonmonotonicity takes place in the families of kinked maps. Furthermore, we construct an open region of kinks for which monotonicity fails. In addition, we give a fundamental reason why our scheme of getting nonmonotonicity of kneading invariants is not possible for maps with negative Schwarzian derivative. 2. Main result. A family of unimodal maps fa is called a family of one-kinked maps if fa = = a · fA, where fA is convex and consists of 4 linear pieces, see [12] (Chapter 9). We assume that fA(0) = fA(1) = 0, fA(c) = 1, where c = 1/2 is the unique critical point of fA and fA is symmetric. The kink A = (x1, y1) ∈ (0, 1/2) × (0, 1) is the point of the graph of fA to the left of the critical point such that fA is not differentiable (Fig. 1). We write the family of one-kinked maps fa : [0, 1] → [0, 1], a ∈ [0, 1] as fa(x) =  s1(a)x if 0 ≤ x ≤ x1, s2(a)(x− c) + a if x1 < x ≤ c, s2(a)(c− x) + a if c < x ≤ 1− x1, s1(a)(1− x) if 1− x1 < x ≤ 1, ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 NONMONOTONICITY OF KNEADING INVARIANTS IN THE FAMILY OF KINKED MAPS 161 � � � � � � � � � � � � �� � � � A C C C C CC @ @ @ c0 1 Fig. 1. The map fA. where s1(a) = a y1 x1 and s2(a) = a 2(1− y1) 1− 2x1 . Also we will denote cn(a) = fn a (c). We prove the following theorem: Theorem 1. If the one-kinked family of maps fa has the following properties: 1) a∗ is such that |f ′a∗(qa∗)| = 1, where qa∗ the is the orientation reversing fixed point of fa∗ ; 2) c2(a∗) < c < qa∗ < c3(a∗) < 1− x1. Then there exists a0 such that for all a ∈ (a∗, a0) K(fa) ≺ K(fa∗). Proof. To prove the theorem we show how the kneading invariant of the one-kinked map K(fa) changes when we change the parameter a. a1 < 1/2 is close to 1/2 then c1(a1) = a1, the slope s2(a1) of the map fa1 is less than 1 and there exists a fixed point x1 < qa1 < 1/2 (i.e., fa1(qa1) = qa1) which is attracting. Therefore K(fa1) = 0∞ (Fig. 2). � � � � � � � � � � � �� S S S PP cqa1 Fig. 2. The map fa1 . a2 = 1/2, then c1(a2) = a2 = 1/2 and there exists a fixed point qa2 = 1/2 and K(fa2) = = C∞ (Fig. 3). � � � � � � � � �� J J J HH qa2 Fig. 3. The map fa2 . ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 162 O. YU. VOLKOVA a3 > a2 is close to a2 such that 1/2 < c1(a3) = a3 < 1−x1, the slope |s2(a3)| < 1 and there exists a fixed point 1/2 < qa3 < 1− x1 which is attracting. Therefore K(fa3) = 1∞ (Fig. 4). � � � � � � � � � � � � �� A A A A HH cqa3 Fig. 4. The map fa3 . a4 is a bit larger than a3 such that c1(a4) = a4 > 1 − x1 and c2 = 1/2 (therefore a4 = = 1/2 + √ 1/4− x1/2y1). Then the slope |s2(a4)| < 1, the fixed point qa4 > 1/2 is attracting and K(fa4) = (1C)∞ (Fig. 5). � � � � � � � � � � � � � � �� B B B B B Z ZZ c1c2 Fig. 5. The map fa4 . a4 < a5 < a∗ is close to a∗. Then since by the assumption of the theorem c2(a∗) < 1/2 < < qa∗ < c3(a∗) < 1 − x1, using continuity argument c2(a5) < 1/2 < qa5 < c3(a5) < 1 − x1. It follows that c4(a5) < qa5 since fa5 is orientation reversing for x > 1/2. Also since |s2(a5)| is a bit less than 1, the fixed point qa5 is attracting and c4(a5) = s2(a5)(1/2 − c3(a5)) + a5 > > c1(a5)− (c3(a5)−1/2) > c1(a5)− (1−x1−1/2) > 1/2. I.e., c2(a5) < 1/2 < c4(a5) < qa5 < < c3(a5) < 1− x1. Therefore K(fa5) = 101∞. a6 = a∗. By the assumptions of the theorem, a fixed point qa∗ > 1/2 is neutral (|f ′a∗(qa∗)| = = 1) and c2(a∗) < 1/2 < qa∗ < c3(a∗) < 1 − x1. This implies that 1/2 < c4(a∗) < qa∗ , c5(a∗) = c3(a∗) and therefore K(fa∗) = 101∞ (Fig. 6). � � � � � � � � � � � �� � � � B B B BB @ @ @ c1c2c c3 q c4 Fig. 6. The map fa∗ . ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 NONMONOTONICITY OF KNEADING INVARIANTS IN THE FAMILY OF KINKED MAPS 163 a7 > a∗, such that a7 is close to a∗. Then c4(a7) lies close to c4(a∗) and qa7 is close to qa∗ , hence c4(a7) < qa7 . So there exists a k0 > 2 such that for 2 < k < k0, 1/2 < c2k(a7) < < c2(k−1)(a7), c2k0−1(a7) > 1/2, c2k0(a7) < 1/2. This means that the kneading invariant K(fa7) = 10111 . . . 10, where the number of 1 before the second 0 is even. Therefore K(fa7) < < K(fa∗) = 101∞. In addition, qa7 > 1/2 is repelling (|f ′a7 (qa7)| > 1). The theorem is proved. Remark 1. Comparing the family fa to the family of quadratic maps Qa(x) = ax(1− x) we have the following: if a < 2 then K(Qa) = 0∞ and qa < 1/2 is attracting; if a = 2 then K(Qa) = C∞ and qa = 1/2; if 2 < a < 3 then K(Qa) = 1∞ and qa > 1/2 is attracting; if a = 3 then K(Qa) = 1∞ and qa > 1/2 is neutrally attracting (|Q′(qa)| = 1); if a > 3 then K(Qa) = 1∞ and qa > 1/2 is repelling (|Q′(qa)| > 1) and there exists a period 2 orbit 1/2 < Q(p) < qa < p; this orbit is attracting; if a = a∗ then K(Qa∗) = 101∞ and qa = Q3 a(1/2) is repelling and there are no attracting periodic points. Lemma 1. Let fa be a unimodal map, c its critical point, Sfa < 0 and suppose there exists a nonrepelling periodic orbit x1, x2, . . . , xn. Then one of the xi has c in its immediate basin. Proof. See [13] (Chapter 1.11). Corollary 1.For a unimodal map fa with Sfa < 0 and a fixed point qa > c there is no parameter a for which K(fa) = 101∞ and qa is nonrepelling. Proof. If such a parameter a exists then, since S(fa) < 0 and qa > c is nonrepelling by Lemma 1, qa will have c in its immediate basin. Therefore all cn must be greater than c. But this is impossible since c2 < c. Example. Let fa be a family of one-kinked maps given above. Then for the open region F = { A = (x1, y1)|0 < x1 < 1/3, 2x2 1 − 2, 5x1 − 0, 25− √ 4x4 1 − 6x3 1 + 3, 25x2 1 − 0, 75x1 + 0, 0625 2x2 1 − x1 − 1 < < y1 < 1 + 5x1 − 2x2 1 + √ 4x4 1 − 12x3 1 + 13x2 1 − 6x1 + 1 2(2− x1) } Theorem 1 holds. The region F corresponds to the conditions qa∗ < c3(a∗), c2(a∗) < 1/2 and c3(a∗) < 1− x1 from Theorem 1. The region F is shown in Fig. 7. Remarks. 2. It is not difficult to see that a similar result holds for a family of maps with more than one kink. Recall that Zdunik considered the case of three-kinked maps. 3. Modifying slightly a result of Taylor [14] one can construct a smooth and convex function F with the same properties as fa∗ . A local geometric argument of Nusse and Yorke [9] shows ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 164 O. YU. VOLKOVA that any smooth map that approximates kinked maps cannot have negative Schwarzian derivati- ve. Lemma 1 and Corollary 1 above give a dynamical argument why this particular procedure of getting nonmonotonicity cannot be realized at all by maps with negative Schwarzian derivative. Fig. 7. The region F . 1. Milnor J., Thurston W. On iterated maps of the interval // Lect. Notes Math. — 1988. — 1342. — P. 465 – 563. 2. Bier M., Bountis T. Remerging Feigenbaum trees in dynamical systems // Phys. Lett. A. — 1984. — 104, № 5. — P. 239 – 244. 3. Inoue M., Kamifukumoto H. Scenarios leading to chaos in forced Lotka – Volterra model // Progr. Theor. Phys. — 1984. — 71. — P. 930 – 937. 4. Tsujii M. A simple proof of monotonicity of entropy in the quadratic family using Ruelle operator. — Banach Centre Preprint, 1997. 5. Tsujii M. A simple proof of monotonicity of entropy in the quadratic family // Ergod. Theory and Dynam. Syst. — 2000. — 20. — P. 925 – 933. 6. Douady A., Hubbard J. H. Publ. Math. Orsay. — 1984 – 1985. 7. Brucks K., Misiurewicz M., and Tresser C. Monotonicity properties of the family of trapezoidal maps // Communs Math. Phys. — 1991. — 137. — P. 1 – 12. 8. Zdunik A. Entropy of transformations of the unit interval // Fund. math. — 1984. — 114. — P. 235 – 241. 9. Nusse H., Yorke J. Period halving for xn+1 = MF (xn) where F has negative Schwarzian derivative // Phys. Lett. A. — 1988. — 127. — P. 328 – 334. 10. Kolyada S. One-parameter families represented by integrals with negative Schwarzian derivative violating monotone bifurcations // Ukr. Mat. Zh. — 1989. — 41, № 2. — P. 258 – 261. 11. Bruin H. Non-monotonicity of entropy of interval maps // Phys. Lett. A. — 1995. — 202. — P. 359 – 362. 12. Brucks K., Bruin H. Topics from one-dimensional dynamics // London Math. Soc. Student Texts 62. — Cambridge Univ. Press, 2004. — 312 p. 13. Devaney R. An introduction to chaotic dynamical systems. — California: Benjamin/Cummings Publ. Co., Menlo Park, 1986. — 336 p. 14. Taylor J. A one-parameter family of unimodal, concave, polynomials maps of the interval exhibiting multi- plicities // Adv. Appl. Math. — 1991. — 12. — P. 464 – 481. Received 25.02.2005 ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2

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