On the Hausdorff dimension of Julia sets of some real polynomials

Levin, Genadi; Zinsmeister, Michel

doi:10.1090/S0002-9939-2013-11660-6

On the Hausdorff dimension of Julia sets of some real polynomials
HTML articles powered by AMS MathViewer

by Genadi Levin and Michel Zinsmeister PDF

Proc. Amer. Math. Soc. 141 (2013), 3565-3572 Request permission

Abstract:

We show that the supremum for $c$ real of the Hausdorff dimension of the Julia set of the polynomial $z\mapsto z^d+c$ ($d$ is an even natural number) is greater than $2d/(d+1)$.

References

Artur Avila and Mikhail Lyubich, Examples of Feigenbaum Julia sets with small Hausdorff dimension, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 71–87. MR 2348955, DOI 10.4171/011-1/3
Adrien Douady, Does a Julia set depend continuously on the polynomial?, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 91–138. MR 1315535, DOI 10.1090/psapm/049/1315535
A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
Genadi Levin and Grzegorz Świa̧tek, Hausdorff dimension of Julia sets of Feigenbaum polynomials with high criticality, Comm. Math. Phys. 258 (2005), no. 1, 135–148. MR 2166843, DOI 10.1007/s00220-005-1332-7
Genadi Levin and Grzegorz Świa̧tek, Measure of the Julia set of the Feigenbaum map with infinite criticality, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 855–875. MR 2643714, DOI 10.1017/S0143385709000340
Jacek Graczyk and Grzegorz Światek, Generic hyperbolicity in the logistic family, Ann. of Math. (2) 146 (1997), no. 1, 1–52. MR 1469316, DOI 10.2307/2951831
R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154. MR 1387085, DOI 10.1112/plms/s3-73.1.105
Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. (2) 147 (1998), no. 2, 225–267. MR 1626737, DOI 10.2307/121009
Michel Zinsmeister, Fleur de Leau-Fatou et dimension de Hausdorff, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 10, 1227–1232 (French, with English and French summaries). MR 1650223, DOI 10.1016/S0764-4442(98)80233-4
O. Kozlovski, W. Shen, and S. van Strien, Density of hyperbolicity in dimension one, Ann. of Math. (2) 166 (2007), no. 1, 145–182. MR 2342693, DOI 10.4007/annals.2007.166.145
Xavier Buff and Arnaud Chéritat, Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive, C. R. Math. Acad. Sci. Paris 341 (2005), no. 11, 669–674 (French, with English and French summaries). MR 2183346, DOI 10.1016/j.crma.2005.10.001
Adrien Douady, Xavier Buff, Robert L. Devaney, and Pierrette Sentenac, Baby Mandelbrot sets are born in cauliflowers, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 19–36. MR 1765083
Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, DOI 10.1007/BF01234434