scholarly journals Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps

2000 ◽  
Vol 75 (4) ◽  
pp. 535-593 ◽  
Author(s):  
C. T. McMullen
Keyword(s):  
Hausdorff Dimension ◽  
Rational Maps ◽  
Geometrically Finite ◽  
Conformal Dynamics

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

scholarly journals Jarník and Julia; a Diophantine analysis for parabolic rational maps for Geometrically Finite Kleinian Groups with Parabolic Elements

2002 ◽  
Vol 91 (1) ◽  
pp. 27 ◽  
Author(s):  
B. O. Stratmann ◽  
M. Urbański
Keyword(s):  
Number Theory ◽  
Multifractal Analysis ◽  
Julia Sets ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Rational Map ◽  
Geometrically Finite ◽  
Conformal Measure ◽  
Further Development

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarník in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a "weak multifractal analysis" of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan's famous dictionary translating between the theory of Kleinian groups and the theory of rational maps.

A framework toward understanding the characterization of holomorphic dynamics

2014 ◽  
Author(s):  
Yunping Jiang
Keyword(s):  
Hyperbolic Geometry ◽  
Spherical Geometry ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Holomorphic Maps ◽  
Critical Set ◽  
Geometrically Finite ◽  
Infinite Degree ◽  
New Framework

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of hyperbolic distortion in hyperbolic geometry, the framework suggested here is based on controlling conformal distortion in spherical geometry. The new framework enables one to relax two of Thurston's assumptions: first, that the iterated map has finite degree and, second, that its post-critical set is finite. Thus, it makes possible to characterize certain rational maps for which the post-critical set is not finite as well as certain classes of entire and meromorphic coverings for which the iterated map has infinite degree.

Geometric measures for parabolic rational maps

1992 ◽  
Vol 12 (1) ◽  
pp. 53-66 ◽  
Author(s):  
M. Denker ◽  
M. Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Rational Maps ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Conformal Measure

AbstractLet h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

scholarly journals Subhyperbolic rational maps on boundaries of hyperbolic components

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yan Gao ◽  
Luxian Yang ◽  
Jinsong Zeng
Keyword(s):  
Critical Points ◽  
Rational Maps ◽  
Rational Map ◽  
Hyperbolic Component ◽  
Quasiconformal Deformation ◽  
Geometrically Finite

<p style='text-indent:20px;'>In this paper, we prove that every quasiconformal deformation of a subhyperbolic rational map on the boundary of a hyperbolic component <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula> still lies on <inline-formula><tex-math id="M2">\begin{document}$ \partial \mathcal{H} $\end{document}</tex-math></inline-formula>. As an application, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components.</p>

Discrete Lyapunov exponents and Hausdorff dimension

2000 ◽  
Vol 20 (1) ◽  
pp. 145-172 ◽  
Author(s):  
SHMUEL FRIEDLAND
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Thermodynamic Formalism ◽  
Sufficient Conditions ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Subshifts Of Finite Type ◽  
Discrete Analogs

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

Sign in / Sign up

Export Citation Format

Share Document