New fibonacci like wild attractors for unimodal interval maps

NEW FIBONACCI-LIKE WILD ATTRACTORS FOR UNIMODAL INTERVAL MAPS ZHANG RONG (B.Sc., Nanjing University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2015 Acknowledgements The last five years have been one of the most important stages in my life. The experience in my Ph.D. period will benefit me for a whole life. I would like to take this opportunity to express my immerse gratitude to all those who have kindly helped me at NUS. At the very first, I am honoured to express my deepest gratitude to my dedicated supervisor, Prof. SHEN Weixiao, who supported me during these five years. The thesis would not have been possible without his great help. He has offered me many great suggestions and ideas with his profound knowledge and rich research experience. From his supervision, I learn the mathematical knowledge and the method of how to mathematical research, both of which will help me a lot for many years. His guidance helped me in all the time of research and writing of this thesis, especially in the fourth year. Moreover, this thesis would not have been possible without the inspiration and support of my supervisor — my thanks and appreciation to him for being part of this journey and making this thesis possible. Without his great help, I am sure that I can not finish my thesis by myself. Without his enthusiasm, encouragement, support and continuous optimism this thesis would hardly have been completed. His guidance into the world of one dimensional dynamics has been a valuable input v vi Acknowledgements for this thesis. He has made available his support in a number of ways, especially towards the completion of this thesis. My great gratitude also goes to my fellow lab mates in NUS: GAO Rui, GAO Bing, DU Zhikun, who have been sharing their insights and research ideas with me in the seminars. Thanks for the simulating discussions, for the sleepless nights we were working together before deadlines. I want to thank them for their unflagging encouragement and serving as role models to me as a junior member of academia. I must thank my fellow graduate friends, who shared the experience at NUS with me and helped me a lot in my daily life. Thanks for accompanying me these years, for always being there when needed. I would like to thank all my friends in Singapore who gave me the necessary distractions from my research and made my stay in Singapore memorable. Completing this thesis would have been all the more difficult were it not for the support and friendship provided by the other graduate students of the Department of Mathematics and Statistics in National University of Singapore. I am indebted to them for their help. Last, but certainly not the least, I would like to thank my family, which creates every possibility for me all these years. Their love provided my inspiration and was my driving force. I owe them everything and wish I could show them just how much I love and appreciate them. Their love and encouragement allowed me to finish this journey. I hope that this work makes you proud. Zhang Rong Jan 2015 Contents Acknowledgements v Summary xi Introduction 1.1 History Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statement of The Results . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real Bound Theorem 2.1 The Fixed Point Equation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Cross Ratio Tool and the Real Koebe Principle . . . . . . . . . . 12 2.3 Combinatorial Properties of the Map f0 . . . . . . . . . . . . . . . . . 13 2.4 Proof of Real Bound Theorem . . . . . . . . . . . . . . . . . . . . . . 19 The Limit Maps 3.1 31 The Limit Map G∞ (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 vii viii Contents 3.2 3.1.1 The upper bound of |w − x0 | if 3.1.2 The Case when the Degree 3.1.3 The Precise Estimation of |w − x0 | . . . . . . . . . . . . . . . 36 is finite . . . . . . . . . . . . 31 → ∞ . . . . . . . . . . . . . . . 34 The Taylor Series of G(x) at x0 and w . . . . . . . . . . . . . . . . . 38 Induced Dynamics and Drift 41 4.1 Induced Dynamics and Its Properties . . . . . . . . . . . . . . . . . . 41 4.2 The Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Proof of the Main Theorem 5.1 5.2 55 Method of Iteration Functions . . . . . . . . . . . . . . . . . . . . . . 55 5.1.1 The Function Φ(x) . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.2 The Function Ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.3 The Length of |ξ0 − x0 | . . . . . . . . . . . . . . . . . . . . . . 65 5.1.4 The Function Υ(x) . . . . . . . . . . . . . . . . . . . . . . . . 71 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 73 Bibliography 74 Appendix 81 A The Construction of Equations 83 A.1 Existence and Properties of Maps with Combinatorial Type (2m + 1, 1) 83 A.2 Topological Properties of the Map with Combinatorial Type (2m + 1, 1) 86 B Bounded Geometry and Renormalization Result 93 B.1 Hyperbolic Geometry and Schwarz Lemma . . . . . . . . . . . . . . . 93 B.2 Bounded Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 B.3 Quasiconformal rigidity of the return maps; renormalization result . . 96 Contents ix C The Maps with Combinatorial Type (3, 1) 113 C.1 Construction of the equation . . . . . . . . . . . . . . . . . . . . . . . 113 C.2 The property of H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 C.2.1 Universal Bound of τ . . . . . . . . . . . . . . . . . . . . . . . 116 C.2.2 Associated Map G(x) . . . . . . . . . . . . . . . . . . . . . . . 117 C.3 The Estimation of |u − x0 | and |H (x0 ) − u| . . . . . . . . . . . . . . 120 τ C.3.1 The Limit Maps H∞ (x) of H (x) . . . . . . . . . . . . . . . . 121 C.3.2 The Lower Bound of |u − x0 | and |H (x0 ) − u| . . . . . . . . . 125 τ C.4 The Taylor Series of G(x) at x0 and u . . . . . . . . . . . . . . . . . . 126 C.3 The Estimation of |u − x0 | and |H (x0 ) − u| 123 τ G∞ (x) = (H τ∞       y=x )2 (x)                             H τ∞ (1) q           q   q q x∞ q τ ∞ x∞ Figure C.7: The Graph of G∞ (x) one fixed point of G−2 on H. Similarly, there is at most one fixed point of G−2 on H. On the real line, G (x) has precisely three fixed points. From Rouch´e’s Theorem, G∞ (x) has at most five fixed points on the real line. i.e. the order 2k + ≤ 5. That means k = or k = 2. The Taylor series of G∞ (x) at x∞ has two cases: G∞ (x) = x + a(x − x∞ )3 + o(|x − x∞ |3 ) or G∞ (x) = x + a(x − x∞ )5 + o(|x − x∞ |5 ) with some a < 0. Claim. The second case is not correct. Proof of the claim. By contradiction, assume G∞ (x) = x + a(x − x∞ )5 + o(|x − x∞ |5 ) for some universal constant a < 0. Then the Taylor series of G(x) at x0 is G(x) = G(x0 ) + (1 − C (x − x0 )5 + o(|x − x0 |5 ) )(x − x0 ) + j=2 with b2 , b3 , b4 → and b5 → a < as → ∞. From above, u, x0 and H (x0 ) are τ fixed points of G(x), let z1 and z2 be another two fixed points of G(x) with z1 ∈ H and z2 ∈ H. In this case, we have z1 → x∞ and z2 → x∞ as → ∞. 124 Chapter C. The Maps with Combinatorial Type (3, 1) From the Taylor series of G(x) at x0 , G(x) = x is equivalent to C − (x − x0 ) + bj (x − x0 )j + b5 (x − x0 )5 + o(|x − x0 |6 ) = 0. j=2 That means z1 , z2 , u and H (x0 ) are four roots of τ − C (x − x0 )j + b5 (x − x0 )5 + o(|x − x0 |5 ) = 0. + j=2 This implies |z1 − x0 | · |z2 − x0 | · |u − x0 | · |H (x0 ) − x0 | = C τ for some universal constant C > 0. From above, C1 C1 |u − x0 | ≤ x0 √ ≤ √ C1 C1 |H (x0 ) − x0 | ≤ x0 √ ≤ √ τ for some universal constant C1 > 0. Hence |z1 − x0 | · |z2 − x0 | = C C2 · ≥ |b5 | · |u − x0 | · |H (x0 ) − x0 | |a| τ for some universal constant C2 > 0. Since a < is also a universal constant, it contradicts the properties |z1 − x0 |, |z2 − x0 | → as → ∞. Therefore, the second case is impossible. We have the following proposition on G∞ (x). Proposition C.5. The limit map G∞ (x) satisfies the following properties: (i) H∞ ◦ G∞ (x) = H∞ (x) τ∞ on (0, x∞ ). (ii) G∞ (x) = H 21 (x) is an increasing map on [0, τ∞ x∞ ]. τ∞ (iii) The Taylor series of G∞ (x) at x∞ is G∞ (x) = x − γ(x − x∞ )3 + o(|x − x∞ |3 ) with γ > 0. C.3 The Estimation of |u − x0 | and |H (x0 ) − u| 125 τ The Lower Bound of |u − x0 | and |H τ1 (x0 ) − u| C.3.2 For simplicity, H(x) denotes H (x). Recall C u ≤ x0 · (1 + √ ) C H (x0 ) ≤ u · (1 + √ ) τ for some universal constant C > and sufficiently large . Moreover, C H (x0 ) ≤ x0 · (1 + √ ) τ for some universal constant C > and sufficiently large . Consider the Taylor series of G(x) at the fixed point x0 , G(x) = G(x0 ) + (1 − C )(x − x0 ) + b2 (x − x0 )2 + b3 (x − x0 )3 + o(|x − x0 |3 ) ⇒ G(x) − x = (x − x0 ) · (− C + b2 (x − x0 )2 + b3 (x − x0 )3 + o(|x − x0 |3 )) where b2 , b3 depend only on . Since G∞ (x) = x + a(x − x∞ )3 + o(|x − x∞ |3 ) with some a < 0, b2 → and b3 → a as → ∞. Since G : (0, τ x0 ) → (H(1/τ ), 1) is in the Epstein class. From Cauchy Theorem, it follows that b2 and b3 are uniformly bounded. i.e. There exists some universal constant M > such that |b2 | ≤ M and |b3 | ≤ M . From above, u and H (x0 ) are fixed points of G(x), then both of them τ satisfy the equations: − − C C + b2 · (u − x0 ) + b3 · (u − x0 )2 + O(|u − x0 |3 ) = 0, + b2 · (H (x0 ) − x0 ) + b3 · (H (x0 ) − x0 )2 + O(|H (x0 ) − x0 |3 ) = τ τ τ Therefore, |u − x0 | · |H (x0 ) − x0 | = τ C |b3 | · ≥ C M· ≥ K0 126 Chapter C. The Maps with Combinatorial Type (3, 1) for some universal constant K0 > and sufficiently large . Since |u−x0 | |x0 | ≤ C √ and τ |H1/τ (x0 )−x0 | |x0 | ≤ C √ , < x0 < u < are uniformly bounded, Therefore, K K |u − x0 | ≥ √ and |H (x0 ) − x0 | ≥ √ τ √ for some universal constant K > 0. That means, |u − x0 |/|x0 | is the order of 1/ . Therefore, K K |u − x0 | ≥ √ and |H (x0 ) − x0 | ≥ √ τ √ for some universal constant K > 0. That means, |u − x0 |/|x0 | is the order of 1/ . C.4 The Taylor Series of G(x) at x0 and u Proposition C.6. The Taylor series of G(x) at u satisfies G(x) − u + + λu (x − u) + bu,2 (x − u)2 − γu (x − u)3 + bu,4 (x − u)4 ≤ C|x − u|5 . The coefficients satisfies the following properties: C C ≤ λu , γu ≤ C, |bu,4 | ≤ √ , |bu,2 | ≤ √ C for some universal constant C > 0. Proof. Step 1. Calculate the derivative of G(x) at u. From above, we know |DG(u)| > 1. The Taylor series of G(x) at u is G(x) = u + DG(u)(x − u) + bu,2 (x − u)2 + bu,3 (x − u)3 + o(|x − u|3 ) where bu,2 → and bu,3 → −γ as → ∞. Since x0 and H (x0 ) are two fixed points τ of G(x), that means both of them are two roots of the equation (DG(u) − 1) + bu,2 (x − u) + bu,3 (x − u)2 + o(|x − u|2 ) = 0. Moreover, |x0 − u| · |H (x0 ) − u| = τ |DG(u) − 1| . |bu,3 | C.4 The Taylor Series of G(x) at x0 and u 127 From above, one has C1 ≤ |x0 − u| · |H (x0 ) − u| ≤ C2 τ for universal constants C1 , C2 > 0. Since DG(u) > 1, one has C1 1+ ≤ DG(u) ≤ + C2 for universal constants C1 , C2 > 0. Step 2. Calculate the upper bound of bu,2 . G(x) = G(u) + (1 + = u + (1 + λu λu )(x − u) + bu,2 (x − u)2 − γu (x − u)3 + O(|x − u|4 ) )(x − u) + bu,2 (x − u)2 − γu (x − u)3 + O(|x − u|4 ) where λ and γ are positive constants. Since G(x) has three fixed points, x0 , u and H (x0 ) with < x0 < u < H (x0 ), we get τ τ G(x0 ) = x0 ⇒ x0 = u + (1 + ⇒ 0= λu λu )(x0 − u) + bu,2 (x0 − u)2 − γu (x0 − u)3 + O(|x0 − u|4 ) + bu,2 (x0 − u) − γu (x0 − u)2 + O(|x0 − u|3 ) Similarly, G(H (x0 )) = H (x0 ) ⇒ = τ τ λu + bu,2 (H (x0 ) − u) − γu (H (x0 ) − u)2 + O(|H (x0 ) − u|3 ) τ τ τ we get bu,2 = (H (x0 ) − u) + (x0 − u) τ γu = |H (x0 ) − u| − |u − x0 | τ = ( |H (x0 ) − u| τ |u − x0 | − 1) · |u − x0 | Therefore, the upper bound of bu is |bu,2 | ≤ |γu | · (|H (x0 ) − u)| + |u − x0 |) ≤ τ C 128 Chapter C. The Maps with Combinatorial Type (3, 1) for some universal constant C > 0. Since H : [x0 , u] → [H (x0 ), u], τ τ H : [u, H (x0 )] → [u, x0 ] τ τ and the Taylor series of G(x) at u, we get |H (x0 ) − u| = (1 + λu τ ≤ (1 + λu ) · |x0 − u| + |bu,2 | · |x0 − u|2 + γ · |x0 − u|3 + O(|x0 − u|4 ) )|x0 − u| + C · C · |x0 − u| + γ · C · |x0 − u| + O ≤ |x0 − u| ≤ 1+ K · |H (x0 ) − u| τ for some universal constant K > 0. That means |H (x0 ) − u| −1 ≤ |x0 − u| −1 |H (x0 ) − u| ≤ τ |x0 − u| and K K τ for some universal constant K > 0. From above, we know bu,2 = |(H (x0 ) − u) + (x0 − u)| = ||H (x0 ) − u| − |u − x0 || τ τ γu |H (x0 ) − u| K C C τ − · |u − x0 | ≤ · = |u − x0 | 2 for some universal constants C > 0. Since γ is a universal constant, we have |bu,2 | ≤ C − 23 since |x0 − u| is order of − and the upper bound of bu,2 from above K 1+ · |x0 − u| for some universal constant K > 0. Similarly, we can get = |x0 − u| for some universal constant C > 0. Therefore, the Taylor series of G(x) at u is G(x) = u + (1 + λu )(x − u) + bu,2 (x − u)2 − γu (x − u)3 + O(|x − u|4 ) C.4 The Taylor Series of G(x) at x0 and u 129 where λu and γu are bounded by universal positive constants. |bu,2 | ≤ C · − 32 for some universal constant C > 0. Step 3. Let us calculate the upper bound of bu,4 and bu,5 . Since G(x) = H 21 (x), τ assume the Taylor series of G(x) and H (x) at u are τ λu G(x) = u + (1 + )(x − u) + bu,2 (x − u)2 − γu (x − u)3 +bu,4 (x − u)4 + bu,5 (x − u)5 + o(|x − u|5 ), and H (x) = u − τ 1+ λu (x − u) + au,2 (x − u)2 + au,3 (x − u)3 +au,4 (x − u)4 + au,5 (x − u)5 + o(|x − u|5 ) respectively. From direct calculation, G(x) = H 21 (x) τ = u + (1 + − 2a2u,2 λu )(x − u) + au,2 1+ λu + au,3 1+ + au,2 a2u,2 + 3au,3 (1 + λu (1 + λu λu )− λu 2+ ) − 2au,3 1+ λu (x − u)2 (x − u)3 1+ λu + au,4 (1 + λu )2 − 1+ λu (x − u)4 +O(|x − u|5 ). Comparing coefficients, the second term satisfies au,2 · (1 + λu )− 1+ λu = |bu,2 | ≤ C , which implies C |au,2 | ≤ √ for some universal constant C > 0. From here, the fourth coefficient satisfies |bu,4 | ≤ au,2 a2u,2 + 3au,3 (1 + λu ) − 2au,3 1+ λu + au,4 (1 + λu )2 − 1+ for some universal constant C > 0, since au,3 and au,4 are uniformly bounded. λu C ≤√ 130 Chapter C. The Maps with Combinatorial Type (3, 1) Corollary C.7. λu C ≤ 3, γu (u − x0 ) − and (H (x0 ) − u) − τ C λu ≤ 3. γu for some universal constant C > 0. Proof. From the Taylor series of G(x) at u, we know 0= λu + bu,2 (x0 − u) − γu (x0 − u)2 + bu,4 (|x0 − u|3 ) + bu,5 (|x0 − u|4 ) ⇒ γu (x0 − u)2 = λu ⇒ γu (x0 − u)2 = λu ⇒ |x0 − u| = − 32 + bu,2 (x0 − u) + bu,4 (|x0 − u|3 ) + O(|x0 − u|4 ) +O λu +O γu , |bu,4 | ≤ C · − 12 C > 0, and |x0 − u| is the order of − 12 since |bu,2 | ≤ C · |H (x0 ) − u| = τ and |bu,5 | ≤ C for some universal constant . Similarly, one has λu +O γu . Corollary C.8. The Taylor Series of G∞ (x) at x∞ is G∞ (x) = x − γ(x − x∞ )3 + b∞,5 (x − x∞ )5 + o(|x − x∞ |5 ) for some γ > and b∞,5 which are bounded by universal constants. Proof. Let → ∞, or precisely take the subsequence conclusion immediately. m → ∞, we can get the C.4 The Taylor Series of G(x) at x0 and u 131 Corollary C.9. The Taylor series of G(x) at x0 satisfies G(x) − x0 + (1 − λx0 )(x − x0 ) + bx0 ,2 (x − x0 )2 − γx0 (x − x0 )3 + bx0 ,4 (x − x0 )4 The coefficients satisfy the following properties: |λx0 − 2λu | ≤ C bx0 ,2 − , γu λu ≤ C , |γx0 − γu | ≤ C C , |bx0 ,4 | ≤ √ for some universal constant C > 0. Moreover, γx0 λx0 C ≤ 2 bx0 ,2 − for some universal constant C > 0. Proof. From above, one has (u − x0 ) − λu C ≤ γu for some universal constant C > 1. Since λu ± O( ), γu x − u = (x − x0 ) − (u − x0 ) = (x − x0 ) − one has u = x0 + (u − x0 ) = x0 + (1 + λu )(x − u) = (1 + = (1 + λu λu λu ± O( ), γu ) (x − x0 ) − )(x − x0 ) − (1 + λu ± O( ) γu λu ) λu ± O( ), γu 2 bu,2 (x − u) = bu,2 (x − x0 ) − λu ± O( ) γu = bu,2 (x − x0 )2 − 2bu,2 λu (x − x0 ) ± O( ) γu ≤ C|x − x0 |5 . 132 Chapter C. The Maps with Combinatorial Type (3, 1) since |bu,2 | ≤ C − 32 for some universal constant C > 0. −γu (x − u)3 = −γu = −γu (x − x0 )3 + + λu λu ± O( ) γu (x − x0 ) − γu λu ± O( ) (x − x0 )2 − 3λu 1 ± O( ) (x − x0 ) λu + O( ), γu λu 1 ) ± O( ) γu λu 1 λu = bu,4 (x − x0 )4 − 4bu,4 ( ) ± O( ) (x − x0 )3 + 6bu,4 ± O( ) (x − x0 )2 γu γu λu λu −4bu,4 ( ) + O( ) (x − x0 ) + bu,4 ( ) + O( ) , γu γu bu,4 (x − u)4 = bu,4 (x − x0 ) − ( λu 1 ) ± O( ) γu 1 λ λu u = bu,5 (x − x0 )5 − 5bu,5 ( ) ± O( ) (x − x0 )4 + 10bu,5 ± O( ) (x − x0 )3 γu γu 1 λu λu ) ± O( ) (x − x0 )2 + 5bu,5 ( ) ± O( ) (x − x0 ) −10bu,5 ( γu γu bu,5 (x − u)5 = bu,5 (x − x0 ) − ( −bu,5 λu γu ±O , Comparing coefficients, one has G(x) = x0 + − 2λu ± O( ) (x − x0 ) + γu λu ± O( ) (x − x0 )2 1 − γu ± O( ) (x − x0 )3 + bu,4 ± O( ) (x − x0 )4 + bu,5 + O( √ ) (x − x0 )5 + o(|x − x0 |5 ) = x0 + (1 − λ x0 )(x − x0 ) + bx0 ,2 (x − x0 )2 − γx0 (x − x0 )3 + bx0 ,4 (x − x0 )4 +bx0 ,5 (x − x0 )5 + o(|x − x0 |5 ) C.4 The Taylor Series of G(x) at x0 and u 133 with the following properties: |λx0 − 2λu | ≤ bx0 ,2 − γu λu ≤ |γx0 − γu | ≤ C C C , , , C |bx0 ,4 | ≤ √ and |bx0 ,5 | ≤ C for some universal constant C > 0. Corollary C.10. The Taylor series of G−1 (x) at u is G−1 (x) = u + (x − u) + cu,2 (x − u)2 + cu,3 (x − u)3 λu 1+ +cu,4 (x − u)4 + cu,5 (x − u)5 + o(|x − u|5 ), where the coefficients satisfy the following conditions: |cu,2 | ≤ |cu,3 − γu | ≤ C C C |cu,4 | ≤ √ and |cu,5 | ≤ C for some universal constant C > 0. Proof. Assume y(x) = (1 + λu )x + bu,2 x2 − γu x3 + bu,4 x4 + bu,5 x5 + o(|x|5 ), and its inverse function is x(y) = y + cu,2 y + cu,3 y + cu,4 y + cu,5 y + o(|y|5 ). λu 1+ 134 Chapter C. The Maps with Combinatorial Type (3, 1) From direct calculation, one has y = (1 + = (1 + +bu,2 −γu λu λu )x + bu,2 x2 − γu x3 + bu,4 x4 + bu,5 x5 + o(|x|5 ) ) y + cu,2 y + cu,3 y + cu,4 y + cu,5 y + o(|y|5 ) λu 1+ y + cu,2 y + cu,3 y + cu,4 y + cu,5 y + o(|y|5 ) + λu y + cu,2 y + cu,3 y + cu,4 y + cu,5 y + o(|y|5 ) λu 1+ +bu,4 y + cu,2 y + cu,3 y + cu,4 y + cu,5 y + o(|y|5 ) + λu +bu,5 y + cu,2 y + cu,3 y + cu,4 y + cu,5 y + o(|y|5 ) λu 1+ + o(|y|5 ), so y = y+ cu,2 · (1 + + cu,3 · (1 + + cu,4 · (1 + +(cu,5 · (1 + + λu λu λu λu )+ )+ bu,2 (1 + λu )2 y2 γu 2bu,2 · cu,2 − λu 1+ (1 + λu )3 ) + bu,2 · (c2u,2 + ) + bu,2 · ( y3 2cu,3 3cu,2 · γu bu,4 )− + λu λu 1+ (1 + ) (1 + λu )4 3c2u,2 2cu,4 3cu,3 + 2c c ) − γ · ( + ) u,2 u,3 u λu λu 1+ 1+ (1 + λu )2 4bu,4 cu,2 bu,5 + )y + o(|y|5 ). λu λu (1 + ) (1 + ) All coefficients of terms with order larger than are zero, one has bu,2 C ⇒ |cu,2 | ≤ , λu (1 + ) γu 2bu,2 · cu,2 C − ⇒ |cu,3 − γu | ≤ , = λu λu (1 + ) (1 + ) cu,2 = − cu,3 y4 and C |cu,4 | ≤ √ and |cu,5 | ≤ C C.4 The Taylor Series of G(x) at x0 and u for some universal constant C > 0. That means the Taylor series of G−1 (x) at u is the form in the corollary. That means G−1 (x) = u + (x − u) + cu,2 (x − u)2 + cu,3 (x − u)3 + λu +cu,4 (x − u)4 + cu,5 (x − u)5 + o(|x − u|5 ) λu ± O( ) (x − u) + cu,2 (x − u)2 + γu ± O( ) (x − u)3 = u+ 1− +cu,4 (x − u)4 + cu,5 (x − u)5 + o(|x − u|5 ). Remark. The unimodal maps with combinatorial type (3, 1) is similar as the Fibonacci circle maps which were studied by Levin and Swiatek in [39], [44], [45]. Using the same method as (2m + 1, 1) with m ≥ 2, we can not prove the drift of unimodal maps with combinatorial type (3, 1) is positive. 135 NEW FIBONACCI-LIKE WILD ATTRACTORS FOR UNIMODAL INTERVAL MAPS ZHANG RONG NATIONAL UNIVERSITY OF SINGAPORE 2015 New Fibonacci-like Wild Attractors for Unimodal Interval Maps Zhang Rong 2015 [...]... existence of wild Cantor attractors of unimodal interval maps It was shown that unimodal interval maps with Fibonacci combinatorics and high criticality have wild attractors by Bruin, Keller, Nowicki and van Strien and the result was later generalized by Bruin to a so-called Fibonacci- like class of maps In this thesis, we provide new examples of unimodal interval maps which possess wild attractors but... 1 Introduction wild attractors for critical circle covering maps with Fibonacci combinatorics and their finding makes the story even more interesting In [45], they introduced a real number ϑ( ) which is called drift such that it is positive if and only if wild Cantor attractors exist They proved that for circle covering maps, lim →∞ ϑ( ) is a finite number while for unimodal Fibonacci maps the limit... using the method in the thesis, we do not show the existence of wild attractors for unimodal interval maps in W2 By similar considerations, we can define associated maps and induced maps for unimodal maps with combinatorial type (3, 1), and the associated map is totally different from others However, the case is similar to the Fibonacci circle maps which are considered in [44] and [45] Second, in order... significantly: lim inf k→∞ Q(k) < ∞ Li and Wang proved that their Fibonacci- like maps have no absolutely continuous invariant probability measure, but left the problem wild attractors wide open The main result of this thesis is that some of the maps in the Li-Wang class have wild attractors In the complex dynamics, the problem of existence of wild attractors is closely related to the problem whether the Julia... an integer for any even integer ≥ 0 (m), 0 (m) such that if the map f is as in the Fact above, then the set {x : f n (x) ∈ U 0 ∪ U 1 for all n ≥ 0} has a positive Lebesgue measure 1.3 Outline of Proof 7 Corollary For any integer m ≥ 2, there exists an integer maps in W2m with even critical order , such that when 0 (m) ≥ and unimodal 0 (m), the set ω(0) is wild Cantor attractors for such maps Remark... [0, 1] → [0, 1] is called non-flat if for any x ∈ [0, 1], Dk f (x) = 0 for some k ≥ 1 Attractors of non-flat interval maps have been one of the main objects in the theory of interval dynamics and studied by Guckenheimer, Blokh, Lyubich, van Strien, Vargas, Martens, among others See [75] for references In particular, a topological attractor A can be one of the following forms: (a) A is a periodic orbit;... (2) for each n ≥ 2, J n = I n and I n−1 ∩ ω(0) ⊆ I n ∪ J n ; (3) for each n ≥ 2, 0 ∈ f Sn (I n ); (4) for each n ≥ 2 and 0 ≤ j < 2m, f Sn +jSn−1 (0) ∈ J n ⊆ I n−1 In particular, the return time of J n to I n−1 is equal to Sn−1 ; (5) for each n ≥ 2, Sn+1 = Sn + 2mSn−1 It is well known that for each m ≥ 1 and each > 1, there is a unimodal map in the class W2m , which has the form x → (λ − 1) − λ|x| For. .. non-decreasing If k−Q(k) is bounded, then f has a wild Cantor attractor when the critical order is sufficiently large enough; else if limk→∞ k−Q(k) = ∞, then f has no wild Cantor attractor As the Fibonacci case corresponds to Q(k) = k − 2, Bruin called his class Fibonacci- like However, in 2014, Li and Wang [50] described some combinatorial types which are extended Fibonacci- like from the viewpoint of generalized... of the main theorem by showing that lim →+∞ ϑ( ) = +∞ In particular, when is large enough, ϑ( ) is positive which implies the existence of wild attractors 1.4 Discussion In this paper, we pay attention to the existence of wild Cantor attractor for unimodal interval maps in W2m , where m ≥ 2 In order to prove the main theorem, we prove the Real Bound Theorem, construct an induced map from the fixed point... Review One of the important facts about iteration of maps on the interval is that the situation is not trivial, for example, the famous theorem ”period 3 implies the existence of periodic points of every period” which was proven by Sharkovskii in 1964 and has been rediscovered by several authors [19] This thesis concerns attractors of unimodal interval maps Let M be a smooth compact manifold and let f . on the existence of wild Cantor attractors of unimodal interval maps. It was shown that unimodal interval maps with Fibonacci combinatorics and high criticality have wild attractors by Bruin,. NEW FIBONACCI- LIKE WILD ATTRACTORS FOR UNIMODAL INTERVAL MAPS ZHANG RONG (B.Sc., Nanjing University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT. later generalized by Bruin to a so-called Fibonacci- like class of maps. In this thesis, we provide new examples of unimodal interval maps which possess wild attractors but are different from the class