- The Spider Algorithm by John H. Hubbard and Dierk Schleicher :
D. A. Brown
- Using spider theory to explore parameter spaces
This is a PhD thesis from Cornell University about family of polynomials P(z)=L(1+z/d)^d, where L is a complex parameter. In this work author is studing location of parameter L for wich polynomial P has an attractive cycle of given length, multiplier and combinatioral type.
Two main tools are used in determining an algorithm for finding these parameters: the well-established theories of external rays in the dynamical and parameter planes and Teichmüller theory. External rays are used to specify hyperbolic components in parameter space of the polynomials and study the combinatorics of the attracting cycle. A properly normalized space of univalent mappings is then employed to determine a linearizing neighborhood of the attracting cycle.
Since the image of a univalent mapping completely determines the mapping, we visualize these maps concretely on the Riemann sphere; with discs for feet and curves as legs connected at infinity, these maps conjure a picture of fat-footed spiders. Isotopy classes of these spiders form a Teichmüller space, and the tools found in Teichmüller theory prove useful in understanding the Spider Space. By defining a contracting holomorphic mapping on this spider space, we can iterate this mapping to a fixed point in Teichmüller space which in turn determines the parameter we seek.
Finally, we extend the results about these polynomial families to the exponential family E(z)=L*e^z. Here, we are able to constructively prove the existence and location of hyperbolic components in the parameter space of E(z). ( text from abstract)
David Brown's Home Page
- David Brown Thurston equivalence without post-critical finitness
- Yuval Fisher page
*Spider* is an XView program which does various things:
* A variant of Thurston's algorithm for computing a postcritically
finite polynomials from the angles of the external rays landing at the critical point.
For example, enter 1/6 and get out c = i, for the quadratic case (If this makes no sense,
nevermind, but notice that the dynamics of 1/6 under multiplication by 2 modulo 1
has some relationship with the orbit of i under z2+i).
* It draws parameter (Mandelbrot set) and dynamical space (Julia sets)
pictures using the Koebe 1/4 theorem as in The Science of Fractal Images.
This part of the code was largely written by Marc Parmet, but it hasn't really seen
the light of day much. This is pretty fast, but I don't really know how fast people draw
things these days.
* It draws external angles on Julia sets.
If you want to understand the relationship between the Mandelbrot set
and the dynamics of Julia sets, this program is for you.
Institute for Non-Linear Science 0402 University of California,
San Diego La Jolla, CA 92093
text from news from: comp.theory.dynamic-sys, sci.fractals Data:1992-11-20 09:55:38 PST
- Dan Erik Krarup Sorensen: The Universal Rabbit
Work from Technical University of Danmark
- Symbolic dynamics, the spider algorithm and finding certain real zeros of polynomials of high degree. T. M. Jonassen .
- Self-similar Groups by Volodymyr Nekrashevych. AMS Bookstore, 2005, ISBN 0821838318, 9780821838310
- A Mathematica Implementation of
Nonlinear Dynamical Systems Theory via the Spider Algorithm and Finding Critical Zeros of High-Degree Polynomials T. M. Jonassen
Author: Adam Majewski
Feel free to e-mail me!