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

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, 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.
Keywords: Julia set, wandering continuum, rational map, Lattès map, torus covering..
Mathematics Subject Classification: 37F10, 37F2.
Citation: Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

G. Cui, W. Peng and L. Tan, Renormalization and wandering curves of rational maps, preprint,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Kiwi, Rational rays and critical portraits of complex polynomials, preprint, 1997/15, SUNY at Stony Brook and IMS.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Levin, On backward stability of holomorphic dynamical systems, Fund. Math., 158 (1998), 97-107.  Google Scholar

[10]

C. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, 135, Princeton University Press, 1994.  Google Scholar

[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.  Google Scholar

[12]

J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.  Google Scholar

[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.  Google Scholar

[14]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Asterisque, 261 (2000), 349-384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

G. Cui, W. Peng and L. Tan, Renormalization and wandering curves of rational maps, preprint,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Kiwi, Rational rays and critical portraits of complex polynomials, preprint, 1997/15, SUNY at Stony Brook and IMS.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Levin, On backward stability of holomorphic dynamical systems, Fund. Math., 158 (1998), 97-107.  Google Scholar

[10]

C. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies, 135, Princeton University Press, 1994.  Google Scholar

[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.  Google Scholar

[12]

J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.  Google Scholar

[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.  Google Scholar

[14]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Asterisque, 261 (2000), 349-384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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