disque de Siegel

CONT: On montre numériquement que pour les applications analytiques du plan complexe avec un disque de Siegel de nombre de rotation irrationnel et un point critique quadratique sur la frontière, la structure d'échelle petite près du point critique peut se décrire asymptomatiquement par une famille à deux paramètres universelle. Résumé français de l'article «Universal small-scale structure near the boundary of Siegel Disks ...» par MacKay et Percival, in Physica, 1987, vo. 26, no. 1-3, pp. 193-202.

CONT: Besides the germination points for buds, the main body of the Mandelbrot set has still other kinds of boundary points ... Unlike in the parabolic case, the boundary does not extend to the fixed point, but the dynamics of the process does not reach it either. The circles shown around the fixed point are invariant circles, i.e. if one takes an initial point on one of the circles, all iterated points will also be on it. Inside the region bounded by the Julia set, the process runs in the following way: first the point springs from smaller, peripheral buds into larger ones until it comes into the disk containing the fixed point. This disk is called a Siegel disk after the German mathematician Carl Ludwig Siegel. When a point has arrived there, it simply rotates around the fixed point on its invariant circle.