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Proof of the Branner-Hubbard conjecture on Cantor Julia sets

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Abstract

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992.

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References

  1. Fatou P. Sur les équations fonctionnelles. Bull Sci Math France, 47: 161–271 (1919); 48: 33-94, 208-314 (1920)

    MathSciNet  Google Scholar 

  2. Julia G. Mémoire sur l’itération des applications fonctionnelles. J Math Pure Appl, 8: 47–245 (1918)

    Google Scholar 

  3. Brolin H. Invariant sets under iteration of rational functions. Ark Mat, 6: 103–144 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  4. Branner B, Hubbard J H. The iteration of cubic polynomials, part II: patterns and parapatterns. Acta Math, 169: 229–325 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hubbard J H. Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. In: Goldberg L R, Phillps A V, eds. Topological Methods in Modern Mathematics. Publish Or Perish, 1993, 467–511

  6. Milnor J. Local connectivity of Julia sets: expository lectures. In: Tan Lei, ed. The Mandelbrot Set, Theme and Variations. London Math Soc Lecture Note Ser No. 274. Cambrige: Cambrige University Press, 2000

    Google Scholar 

  7. Shishikura M. Yoccoz puzzle, t-functions and their applications. Unpublished

  8. Yoccoz J C. On the local connectivity of the Mandelbrot set. Unpublished

  9. Petersen C L. Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math, 177: 163–224 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Petersen C L, Zakeri S. On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann of Math, 159: 1–52 (2004)

    MATH  MathSciNet  Google Scholar 

  11. Roesch P. Topologie locale des méthods de Newton cubiques: plan dynamique. C R Math Acad Sci Paris, 326: 1221–1226 (1998)

    MATH  MathSciNet  Google Scholar 

  12. Roesch P. On local connectivity for the Julia set of rational maps: Newton’s famous example. Ann of Math, 168(1): 127–174 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Levin G, van Strien S. Total disconnectedness and absence of invariant linefields for real polynomials. Asteriques, 261: 161–172 (2000)

    Google Scholar 

  14. Levin G, van Strien S. Bounds for maps of an interval with one critical point of inflection type II. Invent Math, 141: 399–465 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Emerson N D. Dynamics of polynomials with disconnected Julia sets. Discrete Contin Dyn Syst, 9: 801–834 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Milnor J. Dynamics in One Complex Variable. 2nd ed. Braunschweig/Wiesbaden: Vieweg, 2000

    MATH  Google Scholar 

  17. Kozlovski O, van Strien S. Local connectivity and quai-conformal rigidity of non-renormalizable polynomials. Preprint, ArXiv: math.DS/0609710, 2006

  18. McMullen C. Autmorphisms of rational maps. In: Drasin, Earle, Gehring, et al, eds. Holomorphic Functions and Moduli I. New York: Springer, 1998, 30–60

    Google Scholar 

  19. Douady A, Hubbard J H. On the dynamics of polynomial-like mappings. Ann Sci École Norm Sup, 18: 287–344 (1985)

    MATH  MathSciNet  Google Scholar 

  20. Kozlovski O, Shen W, van Strien S. Rigidity for real polynomials. Ann of Math, 165: 749–841 (2007)

    MATH  MathSciNet  Google Scholar 

  21. Yin Y C, Zhai Y. No invariant line fields on Cantor Julia sets. Preprint, ArXiv: math.DS/0609255, 2006; Forum Math, in press

  22. Zhai Y. Ridigity of rational maps with Cantor Julia sets. Sci China Ser A-Math, 51: 79–92 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Roesch P, Yin Y C. The boundary of bounded polynomial Fatou components. C R Math Acad Sci Paris, in press

  24. Cui G Z, Peng W J. On the structure of Fatou domains. Sci in China Ser A-Math, 51: 1167–1186 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kahn J, Lyubich M. The quasi-additivity law in conformal geometry. Ann of Math, in press

  26. Avila A, Kahn J, Lyubich M, et al. Combinatorial rigidity for unicritical polynomials. Ann of Math, in press

  27. Kahn J, Lyubich M. Local connectivity of Julia set for unicritical polynomials. Ann of Math, in press

  28. Tan L, Yin Y C. Unicritical Branner-Hubbard conjecture. Preprint, 2007

  29. Branner B, Hubbard J H. The iteration of cubic polynomials, part I: the global topology of parameter space. Acta Math, 160: 143–206 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  30. Douady A, Hubbard J H. Étude dynamique des polynômes complexes I-II. Publications Mathématiques d’Orsay, 84–02 and 85-04

  31. Ahlfors L. Lectures on quasiconformal Mappings. 2nd ed. University Lecture Series Vol. 38. Providence, RI: American Mathematical Society, 1966

    Google Scholar 

  32. Bruin H, Shen W, van Strien S. Existence of unique SRB-measures is typical for unimodal families. Ann Sci École Norm Sup, 39: 381–414 (2006)

    MATH  Google Scholar 

  33. Lyubich M. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann of Math, 140: 347–404 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lyubich M. Dynamics of quadratic polynomials, I-II. Acta Math, 178: 185–247, 247-297 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  35. Graczyk J, Świątek G. Generic hyperbolicity in the logistic family. Ann of Math, 146: 1–52 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  36. McMullen C. Complex Dynamics and Renormalization. In: Ann of Math Stud No 135. Princeton: Princeton University Press, 1994

    Google Scholar 

  37. Shen W. On the metric properties of multimodal interval maps and C 2 density of Axiom A. Invent Math, 156: 301–403 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Smania D. Complex bounds for multimodal maps: bounded combinatorics. Nonlinearity, 14: 1311–1330 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    WeiYuan Qiu & YongCheng Yin

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  1. WeiYuan Qiu
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  2. YongCheng Yin
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Correspondence to YongCheng Yin.

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This work was supported by the National Natural Science Foundation of China

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Qiu, W., Yin, Y. Proof of the Branner-Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A-Math. 52, 45–65 (2009). https://doi.org/10.1007/s11425-008-0178-9

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  • DOI: https://doi.org/10.1007/s11425-008-0178-9

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