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 constant area ? tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 01/09/2011, 08:34 PM let U be a non-empty simply connected open subset of the complex number plane C which is not all of C , then there exists a biholomorphic mapping f from U onto the open unit disk D. that is riemann's mapping theorem which appears to be important in the field of tetration. now consider iterations of f. we start with a non-empty simply connected open subset of C : U_0. we let U_0 have area on the complex plane equal to A_0. we map U_0 to the open unit disk U_1 by the function f. now we define U_n = f^[n]( U_(n-1) ) and the area on the complex plane for U_n = A_n = area(D) = pi and play with the idea of generalizing to continu iterations : U_r = f^[r]( U_(r-1) ) area( U_r ) = A_r = area(D) = pi clearly this is a pretty strong restriction. i assume this has potential outside of tetration too , such as physics and calculus. in fact , maybe this is already old hat and i forgot about it ( getting too old ? ). it seems alot like things ive seen before ... but not exactly ! intuitively - well mine at least - it seems f^[r] will wander off in 'space' or become periodic. by wandering off in space , i mean that lim n-> oo U_n will go to oo*. ( oo* as in oo on the riemann sphere , possible as a translation of U_x or U_x stretches into direction infinity ) what i mean by periodic is trivial ; f^[r] or equivalently U_r are periodic in r. notice that f^[r] is weaker than f^[z] , we ( ok , I ) only require f^[r] to be real-differentiable in r. ofcourse complex differentiable would be even nicer. oh , before i forget , the trickiest part : there is a bijection between U_0 and U_1. what other bijections exist ? when does U_x biject to U_y ? it seems the radiuses of f^[y-x] , f^[x-y] matter alot. further , i assume we count the overlapping area of U_a and U_b Q times , where Q is the amount of overlap. but for that last , maybe there is a more intresting situation with a different definition. in general a riemann mapping f and its limit n-> oo area (f^[n](U_0)) leads to 0 because of convergeance to fixpoints and oo. but its also intresting to consider non-zero limits of limit n-> oo area (f^[n](U_0)). another remark is that not every taylor series is an f. this may be problematic for many approaches to this problem. too see this consider f = exp(z). but the logaritms of the unit circle gives a twisted riemann surface ! regards tommy1729 « Next Oldest | Next Newest »

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