It has been known that simple dynamical systems can give rise to chaotic behavior, with fractal invariant sets and that in their parameter space, the bifurcation locus (where the dynamics changes) also becomes a fractal set. The self-similarity in the phase space can be explained by the dynamics itself, but the bifurcation locus has a richer structure connected to various dynamical phenomena.

In this lecture, we focus on complex analytic dynamical systems, where the key invariant sets in the phase space are Julia sets and the bifurcation locus in the case of quadratic polynomials is the boundary of the Mandelbrot set.

We will explain various phenomena arising in the bifurcation locus:
The similarity between Julia sets and the Mandelbrot set due to Tan Lei; The self-similarity of the Mandelbrot set which is explained by polynomial-like renormalization due to Douady-Hubbard, Sullivan, McMullen, Lyubich; The discontinuity of Julia sets around parabolic parameters (those with periodic points whose multipliers are roots of unity) due to Douady-Hubbar-Lavaurs; The last property is proved through the analysis of parabolic points, and there are further consequences such as the periodic patterns of the bifurcation locus near the parabolic parameters, and the complexity of the bifurcated Julia sets.
This leads to the proof that the boundary of the Mandelbrot set has Hausdorff dimension two.

We will also discuss on the renormalization theory near the parabolic maps and their applications.