On 3 Mar 2011, at 00:40, Dan Asimov wrote:
David wrote:
For example is it not possible that if we have Rn (n>3) then a non-Mobius mapping/transform in Rn could be such that all non-conformality is restricted to a particular Rx of Rn where x>=1 and x<n-2 ?
This is possible in many ways. E.g.,
(x,y,z) -> (x^2 - y^2, 2xy, z)
is conformal except along the z-axis. (But isn't this the opposite of an example of your previous question?)
--Dan
Those who sleep faster get more rest.
Thanks for all the responses, it seems I missed something as usual - however Dan, that last example doesn't cover the case I suggested since I said Rx of Rn where x<n-2 and in your example x=n-2 - I'm assuming from al the replies that if we had a 4D case then not only is it non-conformal but it must be non-conformal in at least two dimensions, otherwise we'd have an R3 sub-space that was conformal ? Someone else (at http://www.fractalforums.com ) added a little info with respect to Hilbert spaces that also seemed to say that Louiville's theorem holds even for R3 subsets of an Rn space (n>3). However I'll ask the same follow-up question here - given that any complete fractal (to the limit) is essentially a single transcendental transform do the theories/proofs of Louiville and Hilberrt still apply or are they only "proved" for finite transformation functions ? My thinking here is that just because f(z) is not conformal does not necessarily mean that Iter(f(z)) doesn't tend to becoming conformal as we approach the limit.