Advertisement

Inventiones mathematicae

, Volume 212, Issue 2, pp 407–460 | Cite as

A transcendental Julia set of dimension 1

  • Christopher J. Bishop
Article

Abstract

We construct a non-polynomial entire function whose Julia set has finite 1-dimensional spherical measure, and hence Hausdorff dimension 1. In 1975, Baker proved the dimension of such a Julia set must be at least 1, but whether this minimum could be attained has remained open until now. Our example also has packing dimension 1, and is the first transcendental Julia set known to have packing dimension strictly less than 2. It is also the first example with a multiply connected wandering domain where the dynamics can be completely described.

Mathematics Subject Classification

Primary 37F10 Secondary 30D05 37F35 

References

  1. 1.
    Agol, I.: Tameness of hyperbolic 3-manifolds. 2004. Preprint available at arXiv:math/0405568 [math.GT]
  2. 2.
    Avila, A., Buff, X., Chéritat, A.: Siegel disks with smooth boundaries. Acta Math. 193(1), 1–30 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avila, A., Lyubich, M.: Lebesgue measure of Feigenbaum Julia sets. 2015. Preprint available at arXiv:1504.02986 [math.DS]
  4. 4.
    Baker, I .N.: The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser. A I Math 1(2), 277–283 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baker, I.N.: An entire function which has wandering domains. J. Aust. Math. Soc. Ser. A 22(2), 173–176 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baker, I.N.: Some entire functions with multiply-connected wandering domains. Ergod. Theory Dyn. Syst. 5(2), 163–169 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barański, K.: Hausdorff dimension of hairs and ends for entire maps of finite order. Math. Proc. Camb. Philos. Soc. 145(3), 719–737 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baumgartner, M.: Üer Ränder von mehrfach zusammenhängenden wandernden Gebieten. PhD thesis, Christian-Albrechts-Universität zu Kiel (2015)Google Scholar
  9. 9.
    Bergweiler, W.: On the packing dimension of the Julia set and the escaping set of an entire function. Israel J. Math. 192(1), 449–472 (2012a)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bergweiler, W.: On the set where the iterates of an entire function are bounded. Proc. Am. Math. Soc. 140(3), 847–853 (2012b)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bergweiler, W., Hinkkanen, A.: On semiconjugation of entire functions. Math. Proc. Camb. Philos. Soc. 126(3), 565–574 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bergweiler, W., Zheng, J.-H.: On the uniform perfectness of the boundary of multiply connected wandering domains. J. Aust. Math. Soc. 91(3), 289–311 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bers, L.: On boundaries of Teichmüller spaces and on Kleinian groups. I. Ann. Math. 2(91), 570–600 (1970)CrossRefzbMATHGoogle Scholar
  14. 14.
    Bishop, C.J.: True trees are dense. Invent. Math. 197(2), 433–452 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bishop, C.J.: Constructing entire functions by quasiconformal folding. Acta Math. 214(1), 1–60 (2015a)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bishop, C.J.: Models for the Eremenko–Lyubich class. J. Lond. Math. Soc. (2) 92(1), 202–221 (2015b)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bishop, C.J.: The order conjecture fails in S. J. Anal. Math. 127, 283–302 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bishop, C.J.: Models for the Speiser class. Proc. Lond. Math. Soc. (3) 114(5), 765–797 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bishop, C.J., Albrecht, S.: Spesier class Julia sets with dimension near one. (2017). preprintGoogle Scholar
  20. 20.
    Bishop, C.J., Jones, P.W.: Hausdorff dimension and Kleinian groups. Acta Math. 179(1), 1–39 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bishop, C.J., Peres, Y.: Fractals in Probability and Analysis. Cambridge Studies in Advanced Mathematics, vol. 162. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
  22. 22.
    Buff, X., Chéritat, A.: Quadratic Julia sets with positive area. Ann. Math. (2) 176(2), 673–746 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Am. Math. Soc. 19(2), 385–446 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Eremenko, A.E.: On the iteration of entire functions. In: Dynamical Systems and Ergodic Theory (Warsaw, 1986), vol. 23 of Banach Center Publication, pp. 339–345. PWN, Warsaw (1989)Google Scholar
  25. 25.
    Garnett, J.B., Marshall, D.E.: Harmonic Measure, volume 2 of New Mathematical Monographs. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  26. 26.
    Greenberg, L.: Fundamental polyhedra for Keinian groups. Ann. Math. 2(84), 433–441 (1966)CrossRefzbMATHGoogle Scholar
  27. 27.
    Jørgensen, T.: Compact 3-manifolds of constant negative curvature fibering over the circle. Ann. Math. (2) 106(1), 61–72 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Karpińska, B.: Hausdorff dimension of the hairs without endpoints for \(\lambda \exp z\). C. R. Acad. Sci. Paris Sér. I Math. 328(11), 1039–1044 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kisaka, M., Shishikura, M.: On multiply connected wandering domains of entire functions. In: Transcendental dynamics and complex analysis, volume 348 of London Mathematical Society Lecture Note Series, pp. 217–250. Cambridge University Press, Cambridge (2008)Google Scholar
  30. 30.
    Marden, A.: The geometry of finitely generated Kleinian groups. Ann. Math. 2(99), 383–462 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    McMullen, C.T.: Area and Hausdorff dimension of Julia sets of entire functions. Trans. Am. Math. Soc. 300(1), 329–342 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    McMullen, C.T.: Renormalization and 3-manifolds which fiber over the circle, volume 142 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1996)Google Scholar
  33. 33.
    Mihaljević-Brandt, H., Rempe-Gillen, L.: Absence of wandering domains for some real entire functions with bounded singular sets. Math. Ann. 357(4), 1577–1604 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Misiurewicz, M.: On iterates of \(e^{z}\). Ergod. Theory Dyn. Syst. 1(1), 103–106 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Osborne, J.W., Sixsmith, D.J.: On the set where the iterates of an entire function are neither escaping nor bounded. Ann. Acad. Sci. Fenn. Math. 41(2), 561–578 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Przytycki, F.: Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map. Invent. Math. 80(1), 161–179 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rempe, L.: Hyperbolic dimension and radial Julia sets of transcendental functions. Proc. Am. Math. Soc. 137(4), 1411–1420 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rempe-Gillen, L., Sixsmith, D.: Hyperbolic entire functions and the Eremenko–Lyubich class: class B or not class B. (2016). Preprint available at arXiv:1502.00492v2 [math.DS]
  39. 39.
    Rippon, P.: Obituary: Irvine Noel Baker 1932–2001. Bull. Lond. Math. Soc. 37(2), 301–315 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rippon, P .J., Stallard, G .M.: Dimensions of Julia sets of meromorphic functions. J. Lond. Math. Soc. (2) 71(3), 669–683 (2005a)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rippon, P .J., Stallard, G .M.: On questions of Fatou and Eremenko. Proc. Am. Math. Soc 133(4), 1119–1126 (2005b). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Schleicher, D.: Dynamics of entire functions. In: Holomorphic Dynamical Systems, volume 1998 of Lecture Notes in Mathematics, pp. 295–339. Springer, Berlin (2010)Google Scholar
  43. 43.
    Shen, Z., Rempe-Gillen, L.: The exponential map is chaotic: an invitation to transcendental dynamics. Am. Math. Mon. 122(10), 919–940 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. (2) 147(2), 225–267 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stallard, G.M.: The Hausdorff dimension of Julia sets of meromorphic functions. J. Lond. Math. Soc. (2) 49(2), 281–295 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions II. Math. Proc. Camb. Philos. Soc. 119(3), 513–536 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions III. Math. Proc. Camb. Philos. Soc. 122(2), 223–244 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions. IV. J. Lond. Math. Soc. (2) 61(2), 471–488 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Stallard, G.M.: Dimensions of Julia sets of transcendental meromorphic functions. In: Transcendental dynamics and complex analysis, volume 348 of London Mathematical Society Lecture Note Series, pp. 425–446. Cambridge University Press, Cambridge (2008)Google Scholar
  50. 50.
    Sullivan, D.: Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension two. In: Geometry Symposium, Utrecht 1980 (Utrecht, 1980), volume 894 of Lecture Notes in Mathematics, pp. 127–144. Springer, Berlin (1981)Google Scholar
  51. 51.
    Zdunik, A.: Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99(3), 627–649 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentSUNY at Stony BrookStony BrookUSA

Personalised recommendations