Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle

The dual nest for degenerate Yoccoz puzzles

The Yoccoz puzzle is a fundamental tool in Holomorphic Dynamics. The original combinatorial argument by Yoccoz, based on the Branner-Hubbard tableau, counts the preimages of a non-degenerate annulus in the puzzle. However, in some important new applications of the puzzle (notably, matings of quadratic polynomials) there is no non-degenerate annulus. We develop a general combinatorial argument to handle this situation. It allows to derive corollaries, such as the local connectedness of the Julia set, for suitable families of rational maps.

Puzzles and the Fatou-Shishikura injection for rational Newton maps

We establish a principle that we call the Fatou-Shishikura injection for Newton maps of polynomials: there is a dynamically natural injection from the set of non-repelling periodic orbits of any Newton map to the set of its critical orbits. This injection obviously implies the classical Fatou-Shishikura inequality, but it is stronger in the sense that every non-repelling periodic orbit has its own critical orbit. Moreover, for every Newton map we associate a forward invariant graph (a puzzle) which provides a dynamically defined partition of the Riemann sphere into closed topological disks (pu…

余弦函数的有界法ou成分

Bounded Fatou components of cosine functions

研究了余弦函数的有界法ou成分，证明了其在特定条件下的性质及局部连通性。