Abstract:

The local, semi-local, and global dynamics of the complex quadratic polynomials $P_a(z):=e^{2pi i a}z+z^2: C to C$, for irrational values of $a$, have been extensively studied through various methods. The main source of difficulty is the interplay between the tangential movement created by the fixed point and the radial movement caused by the critical point. This naturally brings the arithmetic nature of $a$ into play. Using a renormalization technique developed by H. Inou and M. Shishikura, we analyze this interaction, and in particular, describe the topological and statistical behavior of the orbits of these maps.