American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations
  • DCDS Home
  • This Issue
  • Next Article
    Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions
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 - A, 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.  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.  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.  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.  doi: 10.1007/BF02392534.  Google Scholar

[6]

J. Kiwi, Rational rays and critical portraits of complex polynomials,, preprint, (1997).   Google Scholar

[7]

J. Kiwi, Real laminations and the topological dynamics of complex polynomials,, Adv. in Math., 184 (2004), 207.  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.  doi: 10.1112/plms/pdn055.  Google Scholar

[9]

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

[10]

C. McMullen, Complex Dynamics and Renormalization,, Annals of Mathematics Studies, 135 (1994).   Google Scholar

[11]

C. McMullen, Automorphisms of rational maps,, in Holomorphic Function and Moduli, (1986), 31.  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, (2006), 9.  doi: 10.4171/011-1/1.  Google Scholar

[14]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set,, Asterisque, 261 (2000), 349.   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.  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), (2009), 3.  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.  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.  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.  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.  doi: 10.1007/BF02392534.  Google Scholar

[6]

J. Kiwi, Rational rays and critical portraits of complex polynomials,, preprint, (1997).   Google Scholar

[7]

J. Kiwi, Real laminations and the topological dynamics of complex polynomials,, Adv. in Math., 184 (2004), 207.  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.  doi: 10.1112/plms/pdn055.  Google Scholar

[9]

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

[10]

C. McMullen, Complex Dynamics and Renormalization,, Annals of Mathematics Studies, 135 (1994).   Google Scholar

[11]

C. McMullen, Automorphisms of rational maps,, in Holomorphic Function and Moduli, (1986), 31.  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, (2006), 9.  doi: 10.4171/011-1/1.  Google Scholar

[14]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set,, Asterisque, 261 (2000), 349.   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.  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), (2009), 3.  doi: 10.1201/b10617-3.  Google Scholar

[1]

Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
[2]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[3]

Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
[4]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[5]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[6]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
[7]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[8]

Sébastien Biebler. Lattès maps and the interior of the bifurcation locus. Journal of Modern Dynamics, 2019, 15: 95-130. doi: 10.3934/jmd.2019014
[9]

Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[10]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[11]

Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403
[12]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
[13]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1
[14]

John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723
[15]

Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927
[16]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
[17]

Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940
[18]

Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399
[19]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75
[20]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences