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constant area ?
#1
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
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