Decay of Geometry for a Class of Cubic Polynomials

Abstract

In this paper, we study a class of bimodal cubic polynomials for which its critical points have the same

\omega

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-limit set which is an invariant Cantor set. These maps have generalized Fibonacci combinatorics in terms of generalized renormalization on the twin principal nest. It is proved that such maps possess ‘decay of geometry’ in the sense that the scaling factor of the twin principal nest decreases at least exponentially fast. As an application, we prove that they have no Cantor attractor.

Keywords

Fibonacci / Cubic polynomial / Decay of geometry / Cantor attractor

References

[1] BruinH, KellerG, NowickiT, van StrienS. Wild Cantor attractors exist. Ann. Math., 1996, 143: 97-130

[2] de MeloW, van StrienSOne-dimensional dynamics, 1993BerlinSpringer

[3] GraczykJ, ŚwiatiekGThe real Fatou conjecture, 1998PrincetonPrinceton University Press

[4] GraczykJ, SandsD, ŚwiatiekG. Decay of geometry for unimodal maps: negative Schwarzian case. Ann. Math., 2005, 161: 613-677

[5] Heckman, C.A.: Monotonicity and the construction of quasiconformal conjugacies in the real cubic family. In: Thesis (Ph.D.), Stony Brook, (1996)

[6] JakobsonM, ŚwiatiekG. Metric properties of non-renormalizable S\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S$$\end{document}-unimodal maps. I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$I$$\end{document}. Induced expansion and invariant measures. Ergod. Theory Dyn. Syst., 1994, 14: 721-755

[7] JiH, LiS. On the combinatorics of Fibonacci-like non-renormalizable maps. Comm. Math. Stat., 2020, 8: 473-496

[8] KellerG, NowickiT. Fibonacci maps re(al)-visited. Ergod. Theory Dyn. Syst., 1995, 15: 99-120

[9] KozlovskiO, ShenW, van StrienS. Rigidity for real polynomials. Ann. Math., 2007, 165: 749-841

[10] LehtoO, VirtanenKQuasikonforme Abbildungen, 1965BerlinSpringer

[11] LyubichM. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math., 1994, 140: 347-404

[12] LyubichM, MilnorJ. The Fibonacci unimodal map. J. Am. Math. Soc., 1993, 62425-457

[13] MilnorJ, ThurstonWAlexanderJ. On iterated maps of the interval. Dynamical Systems: Proceedings of the Special Year held at the University of Maryland, College Park, 1986–87, 1988BerlinSpringer465-5631342

[14] ShenW. On the metric properties of multimodal interval maps and C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document} density of Axiom A. Invent. Math., 2004, 1562301-403

[15] ShenW. Decay of geometry for unimodal maps: an elementary proof. Ann. Math., 2006, 163: 383-404

[16] ŚwiatiekG, VargasE. Decay of geometry in the cubic family. Ergod. Theory Dyn. Syst., 1998, 18: 1311-1329

[17] VargasE. Fibonacci bimodal maps. Discrete Contin. Dyn. Syst., 2008, 223807-815

Funding

National Natural Science Foundation of China(12301103)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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