Julia set of z->(exp(z)-1)/2

987 petals

The cauliflower

Quad. poly. with a parabolic fixed point with many petals

Golden mean Siegel disk

Quasiconformal models

Virtual Siegel disks

Pérez-Marco's Riemann sufaces

Siegel disk of exp(z)+c

Digitated Siegel disk

Zakeri's Jordan curve

Pseudo hedgehogs

Near matings

Near tunings

A disconnected rational Julia set

Bifucation loci in parameter spaces

A Julia set equal to the Riemann sphere, looks like Jupiter's moon Callisto

Three representations of the same Julia sets with a Herman ring: z+a*sin(z)+t, with golden mean rotation number.

Parabolic renormalization

Closeup of the Julia set of z->(exp(z)-1)/2 |

High resolution (for inkjet printers) |

A quad. Julia set with 987 parab. petals in approx. Fatou coord. |

Colorful cauliflower |

Checked towel cauliflower filling |

13/34 rot. nb. ind. fix. pt. Julia set. |

(hi res) |

34/89 |

(hi res) |

Golden mean Siegel disk |

(hi res) |

Closeup on the critical point of the golden mean quadratic Siegel disk |

(hi res) |

Approx. domain of def. of the McMullen limit map |

(hi res) |

Ghys model for the quad. gold. mean Sieg. disk. |

(hi res) |

Full Julia set of the previous Blaschke fraction. |

(hi res) |

Virtual Siegel disk in the cauliflower |

(hi res, color) |

(lo res, grayscale) |

Siegel disk tending to the previous |

(hi res) |

Virtual Siegel disk in the 2/5 rabbit |

(hi res) |

Uniformization of Pérez-Marco's tube-log Riemann surface |

(hi res, color) |

(lo res, grayscale) |

(hi res, black and white) |

Uniformization of another Riemann surface of Pérez-Marco |

(hi res, color) |

(lo res, grayscale) |

(hi res, black and white) |

The golden mean fixed Siegel disk in the family exp(z)+c (lo res, grayscale) |

(hi res, black and white) |

Plus some of its invariant circles (medium, color) |

(hi res, color) |

Digitated Siegel disk |

Zakeri's Jordan curve in a slice of the parameter space of cubic polynomials |

A pseudo hedgehog |

Another |

Another |

Another |

Another |

Two mating polynomial Julia sets |

Close-up on the previous one |

Douady's rabbit mating with a dendrite |

Tuning a dragon by a segment |

... closer to the limit |

A disconnected rational Julia set |

Mandelbrot sets everywhere |

A representation of the invariant measure associated to some post-critically finite rational map whose Julia set is the whole sphere. |

Julia set with a Herman ring |

Same Julia set, shades of gray revealing more structure |

Another representation |

Shishikura's invariant class |

exp |

tan |