Robert P. Munafo, 2000 Feb 8.
Certain Julia sets whose parameter C lie on the boundary of a mu-atom but are not bond points exhibit a certain particularly complicated type of dynamics when the orbit of 0+0i is traced. Since these are the iterates of Z at the point C in the Mandelbrot set. Of particular note is the fact that 0+0i neighbors points both inside and outside the Julia Set (as with the Julia sets at bond points) but the outside points are far less numerous, to the extent that in the limit the outside points within a neighborhood of 0+0i have zero measure. Furthermore, all but a measure-zero subset of the points in the Julia set have this property. The result is that the behavior of orbits near the boundary of the Julia set is very difficult to characterize, and there are similar problems characterizing the corresponding behavior near the boundary of the Mandelbrot set.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.
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