The outline of this essay is the study of external rays of Julia Sets of polynomial maps, and then later to extend this on to the Mandelbrot Set. An external ray is, in laymen's terms, the result of drawing a radial line out from the origin and morphing it in such a way that the outside of the unit disc is transformed to the outside of the Julia Set.
Now, I'll explain what a Julia Set is. Consider a function
. Now we can study the effect of the functions 
as 
. Any point which is bounded upon repeated iteration of 
is said to be in the Filled Julia Set 
of . That is, 
. The Julia Set is then said to be 
.The next idea, which will be one of the first of my essay, is the idea of the Böttcher Isomorphism. This is an isomorphism
(Well, in fact the Böttcher isomorphism is in fact 
).We can now study external rays. We can take the lines
and perform the isomorphism 
on them, giving external rays of the Julia Set.Keep watching my blog for further updates to how the essay is going!
2 comments:
I like the ambitious tags for this post :) I'm sure if anyone is searching for blog articles on external rays, you'll be right up there on the top :D
Oh, and the account I'm posting from is an old gaming blog me and my friends used to run
Great. Can you add some information how to compute Boettcher coordinate for julia or mandelbrot set ?
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