284 research outputs found
The Medusa Algorithm for Polynomial Matings
The Medusa algorithm takes as input two postcritically finite quadratic
polynomials and outputs the quadratic rational map which is the mating of the
two polynomials (if it exists). Specifically, the output is a sequence of
approximations for the parameters of the rational map, as well as an image of
its Julia set. Whether these approximations converge is answered using
Thurston's topological characterization of rational maps.
This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by
Christian Henriksen and REU students David Farris, and Kuon Ju Liu. In this
paper we describe the algorithm and its implementation, discuss some output
from the program (including many pictures) and related questions. Specifically,
we include images and a discussion for some shared matings, Lattes examples,
and tuning sequences of matings.Comment: 25 pages, many figures, submitte
Dynamics of Modular Matings
In the paper 'Mating quadratic maps with the modular group II' the current
authors proved that each member of the family of holomorphic
correspondences :
introduced by the first author and C.
Penrose in 'Mating quadratic maps with the modular group', is a mating between
the modular group and a member of the parabolic family of quadratic rational
maps whenever the limit set of is connected.
Here we provide a dynamical description for the correspondences
which parallels the Douady and Hubbard description for
quadratic polynomials. We define a B\"ottcher map and a Green's function for
, and we show how in this setting periodic geodesics play the
role played by external rays for quadratic polynomials. Finally, we prove a
Yoccoz inequality which implies that for the parameter to be in the
connectedness locus of the family , the value of
the log-multiplier of an alpha fixed point which has combinatorial rotation
number lies in a strip whose width goes to zero at rate proportional to
Theorem Proving Using Equational Matings and Rigid E-Unifications
In this paper, it is shown that the method of matings due to Andrews and Bibel can be extended to (first-order) languages with equality. A decidable version of E-unification called rigid E-unification is introduced, and it is shown that the method of equational matings remains complete when used in conjunction with rigid E-unification. Checking that a family of mated sets is an equational mating is equivalent to the following restricted kind of E-unification. Problem: Given â/E = {Ei | 1 ⤠i ⤠n} a family of n finite sets of equations and S = {âŠui, vi⪠| 1 ⤠i ⤠n} a set of n pairs of terms, is there a substitution θ such that, treating each set θ(Ei) as a set of ground equations (i.e. holding the variables in θ(Ei) rigid ), θ(ui) and θ(vi) are provably equal from θ(Ei) for i = 1, ... ,n?
Equivalently, is there a substitution θ such that θ(ui) and θ(vi) can be shown congruent from θ(Ei) by the congruence closure method for i 1, ... , n?
A substitution θ solving the above problem is called a rigid â/E-unifier of S, and a pair (â/E, S) such that S has some rigid â/E-unifier is called an equational premating. It is shown that deciding whether a pair âŠâ/E, S⪠is an equational premating is an NP-complete problem
Unification Procedures in Automated Deduction Methods Based on Matings: A Survey
Unification procedures arising in methods for automated theorem proving based on matings are surveyed. We begin by reviewing some fundamentals of automated deduction, including the Skolem form and the Skolem-Herbrand-GĂśdel theorem. Next, the method of matings for first-order languages without equality due to Andrews and Bibel is presented. Standard unification is described in terms of transformations on systems (following the approach of Martelli and Montanari, anticipated by Herbrand). Some fast unification algorithms are also sketched, in particular, a unification closure algorithm inspired by Paterson and Wegman\u27s method. The method of matings is then extended to languages with equality. This extention leads naturally to a generalization of standard unification called rigid E-unification (due to Gallier, Narendran, Plaisted, and Snyder). The main properties of rigid E-unification, decidability, NP-completeness, and finiteness of complete sets, are discussed
Thurston equivalence for rational maps with clusters
We investigate rational maps with period-one and period-two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is d and the cluster is fixed, the Thurston class of a rational map is fixed by the combinatorial rotation number Ď and the critical displacement δof the cluster cycle. The same result will also be proved in the case where the rational map is quadratic and has a period-two cluster cycle, and we will also show that the statement is no longer true in the higher-degree case
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