## Illinois Journal of Mathematics

### Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$

#### Abstract

We prove that $D_*$, the set of all parameters $\lambda\in\mathbb{C}\setminus\{0\}$ for which the cubic polynomial $f_\lambda$ is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood $D_0$ of the origin 0. Our main result is that if $D_0$ is sufficiently small then the function $D_0\ni\lambda\mapsto\operatorname{HD}(J(f_\lambda))\in\mathbb{R}$ is real-analytic. This function ascribes to the polynomial $f_\lambda$ the Hausdorff dimension of its Julia set $J(f_\lambda)$. The theory of parabolic and hyperbolic graph directed Markov systems with infinite number of edges is used in the proofs.

#### Article information

Source
Illinois J. Math., Volume 55, Number 1 (2011), 157-184.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1355927032

Digital Object Identifier
doi:10.1215/ijm/1355927032

Mathematical Reviews number (MathSciNet)
MR3006684

Zentralblatt MATH identifier
1291.30241

#### Citation

Akter, Hasina; Urbański, Mariusz. Real analyticity of Hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$. Illinois J. Math. 55 (2011), no. 1, 157--184. doi:10.1215/ijm/1355927032. https://projecteuclid.org/euclid.ijm/1355927032

#### References

• R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1980), 11–25.
• M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. Lond. Math. Soc. 43 (1991), 107–118.
• D. Mauldin and M. Urbański, Parabolic iterated function systems, Ergodic Theory Dynam. Systems 20 (2000), 1423–1447.
• D. Mauldin and M. Urbański, Fractal measures for parabolic IFS, Adv. Math. 168 (2002), 225–253.
• D. Mauldin and M. Urbański, Graph directed Markov systems: Geometry and dynamics of limit sets, Cambridge Univ. Press, Cambridge, 2003.
• V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry, Preprint, 2008.
• V. Mayer and M. Urbański, Geometric thermodynamical formalism and real analyticity for meromorphic functions of finite order, Ergodic Theory Dynam. Systems 28 (2008), 915–946.
• V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order, Mem. Amer. Math. Soc. 203 (2010), 1–107.
• F. Przytycki and M. Urbański, Fractals in the plane–-ergodic theory methods, to appear in Cambridge Univ. Press; available at: www.math.unt.edu/~urbanski.
• H. S. Roy and M. Urbański, Analytic families of holomorphic IFS, Nonlinearity 21 (2008), 2255–2279.
• D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99–107.
• H. H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency, Ann. of Math. 168 (2008), 695–748.
• H. H. Rugh, Cones and gauges in complex spaces: Spectral gaps and complex Perron–Frobenius theory, Ann. of Math. (2) 171 (2010), 1707–1752.
• H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, Ergodic Theory Dynam. Systems 30 (2010), 601–633.
• M. Urbański, Analytic families of semihyperbolic generalized polynomial-like mappings, Monatsh. Math. 159 (2010), 133–162.
• M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergodic Theory Dynam. Systems 24 (2004), 279–315.
• M. Urbański and M. Zinsmeister, Geometry of hyperbolic Julia–Lavaurs sets, Indag. Math. 12 (2001), 273–292.