Quadratic matings and ray connections

Abstract

A topological mating is a map defined by gluing together the filled Julia sets of two quadratic polynomials. The identifications are visualized and understood by pinching ray-equivalence classes of the formal mating. For postcritically finite polynomials in non-conjugate limbs of the Mandelbrot set, classical results construct the geometric mating from the formal mating. Here families of examples are discussed, such that all ray-equivalence classes are uniformly bounded trees. Thus the topological mating is obtained directly in geometrically finite and infinite cases. On the other hand, renormalization provides examples of unbounded cyclic ray connections, such that the topological mating is not defined on a Hausdorff space. There is an alternative construction of mating, when at least one polynomial is preperiodic: shift the infinite critical value of the other polynomial to a preperiodic point. Taking homotopic rays, it gives simple examples of shared matings. Sequences with unbounded multiplicity of sharing, and slowly growing preperiod and period, are obtained both in the Chebychev family and for Airplane matings. Using preperiodic polynomials with identifications between the two critical orbits, an example of mating discontinuity is described as well.

arXiv comment32 pages with 6 eps-figures. Attached is a c++ console application for long ray connections

References in corpus (11)

• Thurston's pullback map on the augmented Teichmüller space and applications
• Rational parameter rays of the Mandelbrot set
• Core entropy and biaccessibility of quadratic polynomials
• Core entropy of quadratic polymonials
• Matings with laminations
• Origami, affine maps, and complex dynamics
• Subdivision rule constructions on critically preperiodic quadratic matings
• Constructing rational maps with cluster points using the mating operation
• The Thurston Algorithm for quadratic matings
• Constructing subdivision rules from rational maps
• Shared matings in $V_2$