| 概要 | The local, semi-local, and global dynamics of the complex
quadratic polynomials
:=e^{2\pi i \alpha}z+z^2: \mathbb{C}\to
<br>\mathbb{C})
, for irrational values of  , 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 into play.
Using a renormalization technique developed by H. Inou and M. Shishikura,
we analyze this interaction, and in particular, describe the topological
behavior of the orbit of typical points under these maps. |