Problems on Local Connectivity. 1

with a xed point of multiplier = e 2 i at the origin. Take to be real and irrational. For generic choice of (in the sense of Baire category), Cremer showed that there is no local linearizing coordinate near the origin. We will say brie y that the origin is a Cremer point, or that P is a Cremer polynomial . According to Sullivan and Douady, the existence of such a Cremer point implies that the Julia set is not locally connected. More explicitly, let t( ) be the angle of the unique external ray which lands at the corresponding point of the Mandelbrot set. For generic choice of , Douady has shown that the corresponding ray in the dynamic plane does not land, but rather has an entire continuum of limit points in the Julia set. (Compare [S ].) Furthermore, the t( )=2 ray in the dynamic plane accumulates both at the xed point 0 and its pre-image .