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.

The geometry of conformal measures for parabolic rational maps

2000 ◽  
Vol 128 (1) ◽  
pp. 141-156 ◽  
Author(s):  
B. O. STRATMANN ◽  
M. URBAŃSKI
Keyword(s):  
Julia Set ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Rational Map ◽  
Uniformly Perfect ◽  
Gauge Functions ◽  
Efficient Control ◽  
Rényi Dimension ◽  
Conformal Measure ◽  
Conformal Measures

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

Density of hyperbolicity for rational maps with Cantor Julia sets

2011 ◽  
Vol 32 (5) ◽  
pp. 1711-1726 ◽  
Author(s):  
WENJUAN PENG ◽  
YONGCHENG YIN ◽  
YU ZHAI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

An exceptional set in the ergodic theory of rational maps of the Riemann sphere

1997 ◽  
Vol 17 (2) ◽  
pp. 253-267 ◽  
Author(s):  
A. G. ABERCROMBIE ◽  
R. NAIR
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Hausdorff Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Maps ◽  
Exceptional Set ◽  
Rational Map ◽  
Pointwise Ergodic Theorem ◽  
Conformal Measure

A rational map $T$ of degree not less than two is known to preserve a measure, called the conformal measure, equivalent to the Hausdorff measure of the same dimension as its Julia set $J$ and supported there, with respect to which it is ergodic and even exact. As a consequence of Birkhoff's pointwise ergodic theorem almost every $z$ in $J$ with respect to the conformal measure has an orbit that is asymptotically distributed on $J$ with respect to this measure. As a counterpoint to this, the following result is established in this paper. Let $\Omega(z)=\Omega_{T}(z)$ denote the closure of the set $\{T^{n}(z):n=1,2,\ldots\}$. For any expanding rational map $T$ of degree at least two we set \[ S(z_{0})=\{z\in J:z_{0}\not\in \Omega_{T}(z)\}. \] We show that for all $z_{0}$ the Hausdorff dimensions of $S(z)$ and $J$ are equal.

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)) = ∞.

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>

Limiting behavior of Julia sets of singularly perturbed rational maps

2014 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Singular Perturbations ◽  
Julia Sets ◽  
Singularly Perturbed ◽  
Rational Maps ◽  
Rational Map ◽  
Limiting Behavior ◽  
Jordan Curves ◽  
Hyperbolic Component ◽  
Dynamical Properties ◽  
Sierpinski Gaskets

This chapter surveys dynamical properties of the families fsubscript c,𝜆(z) = zⁿ + c + λ‎/zᵈ for n ≥ 2, d ≥ 1, with c corresponding to the center of a hyperbolic component of the Multibrot set. These rational maps produce a variety of interesting Julia sets, including Sierpinski carpets and Sierpinski gaskets, as well as laminations by Jordan curves. The chapter describes a curious “implosion” of the Julia sets as a polynomial psubscript c = zⁿ + c is perturbed to a rational map fsubscript c,𝜆. In this way the chapter shows yet another way of producing rational maps through “singular” perturbations of complex polynomials.

scholarly journals Rational maps whose Julia sets are Cantor circles

2013 ◽  
Vol 35 (2) ◽  
pp. 499-529 ◽  
Author(s):  
WEIYUAN QIU ◽  
FEI YANG ◽  
YONGCHENG YIN
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Cantor Set ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

scholarly journals On (non-)local connectivity of some Julia sets

2014 ◽  
Author(s):  
Alexandre Dezotti ◽  
Pascale Roesch
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complete Characterization ◽  
Rational Maps ◽  
Local Connectivity ◽  
Rational Map ◽  
Critical Set ◽  
Quadratic Polynomials ◽  
Non Local

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.

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