Gallery
A movie for exponential convergence of circle packings.
Julia set/limit set that are quasiconformal to circle packings
Self-bump on the main hyperbolic component
Coorespondence between graphs, reflection groups and anti-rational maps
2 connected simple planar graph
Limit set of a kissing reflection group
Julia set of a critically fixed anti-rational map
Julia set of critically fixed anti-rational maps associated to Plantonic solids
Tetrahedron
Cube
Octahedron
Icosahedron
Dodecahedron
J-conjugate hyperbolic components
The Julia sets of two hyperbolic rational maps that are conjugate on their Julia sets, but are in different hyperbolic components. The rational map is of the form as above for sufficiently small t. There are two choices of cubic roots which gives the identification of the formula with Z_3 \times Z_3.
(0,0) is in the same hyperbolic component as (1,1) by moving t around 0 once. This turns out to be the only additional identification, thus the formula gives 3 J-conjugate hyperbolic components. One can see the Julia components are twisted in a different (and incompatible) rate.
Buried Sierpinski Carpet and Tree mapping schemes
Buried Sierpinski Carpet
The Julia set of a hyperbolic rational map with finitely connected Fatou sets. It contains a fixed Sierpinski carpet as a buried component.
Associated Tree mapping scheme
The tree mapping scheme associated to the buried Sierpinski carpet example.
A 'nested mating' of z^4-1 and z^4-1.10658-0.24848i.
A 'nested mating' of a basilica and a rabbit.
A movie from Cantor set of circle to mono nested mating of a basilica.
