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I have been puzzled by this exercise for some time. I won't repeat it here as the hint refers to other exercises, so I would be copying a whole page of the book.

My question is as follows. After having defined the sets [tex]S_i[/tex] and picked a complex number [tex]s[/tex] not belonging to any of the [tex]S_i[/tex] it is trivial to use part (b) of exercise 55 to show that for some large [tex]k[/tex] we have [tex]M_i^k(s)\in S_i[/tex] for all [tex]i[/tex].

Now the problem is that we still need to prove that [tex]M_i^{nk}(s)\in S_i[/tex] for all [tex]n\neq 0[/tex] i.e. all the exponents, but I don't see how this can be done for negative exponents.

Further, I have no idea how to prove that there exists such a [tex]k[/tex] that [tex]M_i^k(S_j)\subset S_i[/tex] for all [tex]j\neq i[/tex] and further for all the exponents of [tex]M_i^k[/tex]. It doesn't even seem plausible that this could happen as any element close to [tex]w'_i[/tex] is (for a large enough [tex]k[/tex]) mapped into [tex]U_i[/tex] i.e. close to [tex]w_i[/tex] by part (b) of the previous exercise. Thus, it would seem that choosing some [tex]z\in U_i'[/tex] closer and closer to [tex]w_i'[/tex] would require [tex]k[/tex] to grove without bound...

I'm just wondering whether anyone has ever done this exercise or has some tips on how to approach it?

Thanks.