Wandering continua for rational maps

Guizhen Cui; Yan  Gao

doi:10.3934/dcds.2016.36.1321

Article Contents

2016, Volume 36, Issue 3: 1321-1329. Doi: 10.3934/dcds.2016.36.1321

This issue Previous Article Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions Next Article Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations

Wandering continua for rational maps

Guizhen Cui^1, and
Yan Gao^2,

1.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190
2.
Mathemaitcal School of Sichuan University, ChengDu, 610065

Received: July 31, 2014

Revised: May 31, 2015

Published: May 2017

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Abstract

We prove that a Lattès map admits an always full wandering continuum if and only if it is flexible. The full wandering continuum is a line segment in a bi-infinite or one-side-infinite geodesic under the flat metric.

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Mathematics Subject Classification: 37F10, 37F20.

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References

[1]	A. Blokh and G. Levin, An inequality for laminations, Julia sets and 'growing trees', Erg. Th. and Dyn. Sys., 22 (2002), 63-97.doi: 10.1017/S0143385702000032.
[2]	B. Branner and J. Hubbard, The iteration of cubic polynomials. Part II. Patterns and parapatterns, Acta Mathematica, 169 (1992), 229-325.doi: 10.1007/BF02392761.
[3]	G. Cui, W. Peng and L. Tan, Renormalization and wandering curves of rational maps, preprint, arXiv:1403.5024.
[4]	G. Cui and L. Tan, A characterization of hyperbolic rational maps, Invent. Math., 183 (2011), 451-516.doi: 10.1007/s00222-010-0281-8.
[5]	A. Douady and J. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297.doi: 10.1007/BF02392534.
[6]	J. Kiwi, Rational rays and critical portraits of complex polynomials, preprint, 1997/15, SUNY at Stony Brook and IMS.
[7]	J. Kiwi, Real laminations and the topological dynamics of complex polynomials, Adv. in Math., 184 (2004), 207-267.doi: 10.1016/S0001-8708(03)00144-0.
[8]	O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. London Math. Soc., 99 (2009), 275-296.doi: 10.1112/plms/pdn055.
[9]	G. Levin, On backward stability of holomorphic dynamical systems, Fund. Math., 158 (1998), 97-107.
[10]	C. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, 135, Princeton University Press, 1994.
[11]	C. McMullen, Automorphisms of rational maps, in Holomorphic Function and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60.doi: 10.1007/978-1-4613-9602-4_3.
[12]	J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.
[13]	J. Milnor, On Lattès Maps, in Dynamics on the Riemann Spheres, A Bodil Branner festschrift (eds. P. G. Hjorth and C. L. Petersen), European Mathematical Society, 2006, 9-43.doi: 10.4171/011-1/1.
[14]	K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Asterisque, 261 (2000), 349-384.
[15]	W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Science in China Series A, 52 (2009), 45-65.doi: 10.1007/s11425-008-0178-9.
[16]	W. Thurston, The combinatorics of iterated rational maps, in Complex Dynamics: Families and Friends (ed. D. Schleicher), A K Peters/CRC Press, 2009, 3-130.doi: 10.1201/b10617-3.