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March  2016, 36(3): 1321-1329. doi: 10.3934/dcds.2016.36.1321

Wandering continua for rational maps

1. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190

2. 

Mathemaitcal School of Sichuan University, ChengDu, 610065, China

Received  August 2014 Revised  June 2015 Published  August 2015

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.
Citation: Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321
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.

show all references

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.

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