Hausdorff dimension and conformal dynamics, III: Computation of dimension

<abstract abstract-type="TeX"><p>This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials <i>f<sub>c</sub></i>(<i>z</i>) = <i>z</i><sup>2</sup>+<i>c, c</i> ∈ [-1, 1/2]; and (c) the family of rational maps <i>f<sub>t</sub></i>(<i>z</i>) = <i>z</i>/<i>t</i> + 1/<i>z, t</i> ∈ (0, 1]. We also calculate H.dim (Λ) ≈ 1.305688 for the Apollonian gasket, and H.dim (<i>J</i>(<i>f</i>)) ≈ 1.3934 for Douady's rabbit, where <i>f</i>(<i>z</i>) = <i>z</i><sup>2</sup> + <i>c</i> satisfies <i>f</i><sup>3</sup>(0) = 0.

[1]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[2]  David Ruelle.  Repellers for real analytic maps , 1982 .

[3]  R. Phillips,et al.  The laplacian for domains in hyperbolic space and limit sets of Kleinian groups , 1985 .

[4]  Michel Zinsmeister.  Formalisme thermodynamique et systèmes dynamiques holomorphes , 1996 .

[5]  Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps , 2000 .

[6]  A. Beardon.  The Exponent of Convergence of Poincaré Series , 1968 .

[7]  André C. Rocha,et al.  Analyticity of Hausdorff dimension of limit sets of Kleinian groups , 1997 .

[8]  S. Shenker,et al.  Strange objects in the complex plane , 1983 .

[9]  Tong Zhang,et al.  Densities of Self-Similar Measures on the Line , 1995, Exp. Math..

[10]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[11]  Lei Tan,et al.  Local connectivity of the Julia set for geometrically finite rational maps , 1996 .

[12]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[13]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[14]  R. Gardner,et al.  On the Hausdorff dimension of a set of complex continued fractions , 1983 .

[15]  R. Bowen.  Hausdorff dimension of quasi-circles , 1979 .

[16]  D. Boyd.  The residual set dimension of the Apollonian packing , 1973 .

[17]  Laminations in holomorphic dynamics , 1994, math/9412233.

[18]  J. Milnor,et al.  Dynamics in One Complex Variable: Introductory Lectures , 2000 .

[19]  D. Sullivan.  Related aspects of positivity in Riemannian geometry , 1987 .

[20]  A. Douady,et al.  Étude dynamique des polynômes complexes , 1984 .

[21]  Michel Zinsmeister,et al.  Implosion parabolique et dimension de Hausdorff , 1997 .

[22]  Curtis T. McMullen,et al.  Complex Dynamics and Renormalization , 1994 .

[23]  C. McMullen.  Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups , 1999 .

[24]  Michel Zinsmeister,et al.  Quelques résultats sur la dimension de Hausdorff des ensembles de Julia des polynômes quadratiques , 1996 .

[25]  D. Sullivan,et al.  Estimating small eigenvalues of Riemann surfaces , 1987 .

[26]  P. Doyle.  On the bass note of a Schottky group , 1988 .

[27]  C. McMullen.  Hausdorr Dimension and Conformal Dynamics Ii: Geometrically Nite Rational Maps , 1998 .

[28]  D. Sullivan.  Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups , 1984 .