THE RENORMALIZATION METHOD

AND QUADRATIC-LIKE MAPS

Yunping Jiang

Abstract. The renormalization of a quadratic-like map is studied. The threedimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the a priori complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan’s sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the a priori complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so that for every point of the subset, the corresponding quadratic polynomial is unbranched and has the a priori complex bounds.

Typeset by AMS-TEX