## Mandelbrot set and quadratic polynomials

The Mandelbrot set*M*is the connectedness locus of complex quadratic polynomials f

_{c}(z) = z

^{2}+ c . The filled Julia set

*K*

_{c}contains those values z that are not going to ∞ under iteration. The following videos illustrate bifurcations of

*K*

_{c}or self-similarity properties of

*M*, respectively. See Mandel demos 2, 5, 6, 7 for a more extensive explanation.

### Bifurcation of Julia sets

Approaching the airplane root with Fatou-Lavaurs translation (1/σ ~ time): play — show.Airplane primitive parabolic implosion (c ~ time, 1/σ accelerated): play — show.

Rabbit satellite bifurcation: play — show.

Bifurcation of preperiodic points and rays at c = γ

_{M}(5/12) : play — show.

### Similarity and self-similarity of the Mandelbrot set

Slideshow of Feigenbaum scaling: play — show.Slideshow of rescaled limbs converging to the Lavaurs elephant: play — show.

Embedded Julia set similar to Misiurewicz Julia set: play — show (24 MB).

Embedded Julia set similar to Siegel Julia set: play — show (30 MB).

Nodes converging to a Misiurewicz point within an embedded Julia set: play — show (42 MB).

Nested structures at a node, converging to another Misiurewicz point: play — show (43 MB).

Local similarity, zooming into both planes, dynamic plane rescaled: play — show (24 MB).

Local similarity, the rescaled Julia set is changing with the parameter c: play — show.

Renormalization, the Julia set is changing with the parameter c without rescaling: play — show (24 MB).

Slideshow of asymptotic similarity at a = γ

_{M}(9/56) with the scale γ=1: play — show.

Slideshow of asymptotic similarity at a = γ

_{M}(9/56) with the scale γ=3/2: play — show.

Slideshow of asymptotic similarity at a = γ

_{M}(5/12) with the scale γ=2: play — show.

### Visualization of the Thurston algorithm

Recapture turns Basilica into Airplane, one step per second: play — show.Left Dehn twist turns Rabbit into Airplane, one step per second: play — show.

Right Dehn twist turns Rabbit into Corabbit, one step per second: play — show.

Eigenvalues of the Thurston pullback or pushforward are shown in a slideshow, such that the color indicates the quadrant of the value of the characteristic polynomial on the square [-1, 1] × [-i, i]. Compare this paper by Xavier Buff, Adam L. Epstein, and Sarah Koch.

For two sequences of centers with periods 5, 7, ..., 35 converging geometrically to the Misurewicz point with angle 5/12, most eigenvalues accumulate on a small circle, and the leading eigenvalue converges as well: play — show.

For a sequence of centers with periods 5, 8, ..., 62 converging to the Airplane root with angle 3/7, eigenvalues accumulate on the unit circle: play — show.