On the derivative of the Hausdorff dimension of the quadratic Julia sets

Jaksztas, Ludwik

doi:10.1090/S0002-9947-2011-05208-6

On the derivative of the Hausdorff dimension of the quadratic Julia sets
HTML articles powered by AMS MathViewer

by Ludwik Jaksztas PDF

Trans. Amer. Math. Soc. 363 (2011), 5251-5291 Request permission

Abstract:

Let $d(c)$ denote the Hausdorff dimension of the Julia set $J_c$ of the polynomial $f_c(z)=z^2+c$. The function $c\mapsto d(c)$ is real-analytic on the interval $(-3/4,1/4)$, which is included in the main cardioid of the Mandelbrot set. It was shown by G. Havard and M. Zinsmeister that the derivative $d’(c)$ tends to $+\infty$ as fast as $(1/4-c)^{d(1/4)-3/2}$ when $c\nearrow 1/4$. Under numerically verified assumption $d(-3/4)<4/3$, we prove that $d’(c)$ tends to $-\infty$ as $-(c+3/4)^{3d(-3/4)/2-2}$ when $c\searrow -3/4$.

References

Jon Aaronson, Manfred Denker, and Mariusz Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 495–548. MR 1107025, DOI 10.1090/S0002-9947-1993-1107025-2
Xavier Buff and Lei Tan, Dynamical convergence and polynomial vector fields, J. Differential Geom. 77 (2007), no. 1, 1–41. MR 2344353
Manfred Denker and Mariusz Urbański, Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), no. 6, 561–579. MR 1129999, DOI 10.1515/form.1991.3.561
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
Adrien Douady, Pierrette Sentenac, and Michel Zinsmeister, Implosion parabolique et dimension de Hausdorff, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 7, 765–772 (French, with English and French summaries). MR 1483715, DOI 10.1016/S0764-4442(97)80057-2
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
G. Havard, M. Zinsmeister, Le chou-fleur a une dimension de Hausdorff inférieure à 1,295, Preprint, 2000.
Guillaume Havard and Michel Zinsmeister, Thermodynamic formalism and variations of the Hausdorff dimension of quadratic Julia sets, Comm. Math. Phys. 210 (2000), no. 1, 225–247. MR 1748176, DOI 10.1007/s002200050778
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
Feliks Przytycki and Mariusz Urbański, Conformal fractals: ergodic theory methods, London Mathematical Society Lecture Note Series, vol. 371, Cambridge University Press, Cambridge, 2010. MR 2656475, DOI 10.1017/CBO9781139193184
Curtis T. McMullen, Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), no. 4, 535–593. MR 1789177, DOI 10.1007/s000140050140
Curtis T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math. 120 (1998), no. 4, 691–721. MR 1637951, DOI 10.1353/ajm.1998.0031
David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
F. Schweiger, Numbertheoretical endomorphisms with $\sigma$-finite invariant measure, Israel J. Math. 21 (1975), no. 4, 308–318. MR 384735, DOI 10.1007/BF02757992
Michel Zinsmeister, Formalisme thermodynamique et systèmes dynamiques holomorphes, Panoramas et Synthèses [Panoramas and Syntheses], vol. 4, Société Mathématique de France, Paris, 1996 (French, with English and French summaries). MR 1462079