scholarly journals Conformal measures and matings between Kleinian groups and quadratic polynomials

2007 ◽  
Vol 193 (2) ◽  
pp. 95-132
Author(s):  
Marianne Freiberger
Keyword(s):  
Kleinian Groups ◽  
Quadratic Polynomials ◽  
Conformal Measures

No elliptic limits for quadratic maps

1999 ◽  
Vol 19 (1) ◽  
pp. 127-141 ◽  
Author(s):  
CARSTEN LUNDE PETERSEN
Keyword(s):  
Periodic Orbits ◽  
Rotation Number ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Holomorphic Maps ◽  
Quadratic Polynomials ◽  
Quadratic Maps ◽  
New Construction ◽  
Principal Tool ◽  
Quasiconformal Deformations

We establish bounds for the multipliers of those periodic orbits of $R_\mu(z) = z(z+\mu)/(1+\overline\mu z) $, which have a Poincaré rotation number $ p/q $. The bounds are given in terms of $ p/q $ and the (logarithmic) hororadius of $\mu$ to $e^{2\pi ip/q} $. The principal tool is a new construction denoted a ‘star’ of an immediate attracting basin. The bounds are used to prove properties of the space of Möbius conjugacy classes of quadratic rational maps. These properties are related to the mating and non-mating conjecture for quadratic polynomials lsqb;Ta]. Moreover they are also reminiscent of Chuckrows theorem on the non-existence of elliptic limits of loxodromic elements in quasiconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomorphic 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.

scholarly journals Distribution of polynomials with cycles of a given multiplier

2011 ◽  
Vol 201 ◽  
pp. 23-43 ◽  
Author(s):  
Giovanni Bassanelli ◽  
François Berteloot
Keyword(s):  
Quadratic Polynomials ◽  
Bifurcation Locus

AbstractIn the space of degreedpolynomials, the hypersurfaces defined by the existence of a cycle of periodnand multipliereiθare known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

Kleinian groups acting onS 3 which are extensions

1998 ◽  
Vol 3 (4) ◽  
pp. 355-374
Author(s):  
L. Potyagailo
Keyword(s):  
Kleinian Groups
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