Article Contents
Article Contents

Wandering continua for rational maps

• 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.
Mathematics Subject Classification: 37F10, 37F20.

 Citation:

•  [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.