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Any counterexample to Makienko’s conjecture is an indecomposable continuum

  • CLINTON P. CURRY (a1), JOHN C. MAYER (a1), JONATHAN MEDDAUGH (a2) and JAMES T. ROGERS Jr (a2)
Abstract
Abstract

Makienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ→ℂ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

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[1]A. F. Beardon . Iteration of Rational Functions. Springer, Berlin, 1991.

[2]D. K. Childers , J. C. Mayer and J. T. Rogers Jr . Indecomposable continua and the Julia sets of polynomials. II. Topology Appl. 153(10) (2006), 15931602.

[3]D. K. Childers , J. C. Mayer , H. Murat Tuncali and E. D. Tymchatyn . Indecomposable continua and the Julia sets of rational maps. Complex Dynamics (Contemporary Mathematics, 396). American Mathematical Society, Providence, RI, 2006, pp. 120.

[5]R. L. Devaney , D. M. Look and D. Uminsky . The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54(6) (2005), 16211634.

[8]J. C. Mayer and J. T. Rogers Jr . Indecomposable continua and the Julia sets of polynomials. Proc. Amer. Math. Soc. 117(3) (1993), 795802.

[9]C. McMullen . Automorphism of rational maps. Holomorphic Functions and Moduli, I (Mathematical Sciences Research Institute Publications, 10). Springer, New York, 1988,pp. 3160.

[10]J. Milnor and T. Lei . A “Sierpiński carpet” as Julia set. Appendix F in Geometry and dynamics of quadratic rational maps. Experiment. Math. 2(1) (1993), 3783.

[11]S. Morosawa . On the residual Julia sets of rational functions. Ergod. Th. & Dynam. Sys. 17(1) (1997), 205210.

[12]S. Morosawa . Julia sets of subhyperbolic rational functions. Complex Var. Theory Appl. 41(2) (2000), 151162.

[14]J. T. Rogers Jr . Diophantine conditions imply critical points on the boundaries of Siegel disks of polynomials. Comm. Math. Phys. 195(1) (1998), 175193.

[15]J. Qiao . Topological complexity of Julia sets. Sci. China Ser. A 40(11) (1997), 11581165.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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