Re: Corners in metric spaces

I have a kind request of help for you Mr. Sheskey
(or do you happen to know somebody) or some other
reader of the thread. I knew that the sentences
1 and sentences 2 should be true on all or most cases
but now I think I have may found another good reason why
it is important. I would even more appreciate the
refutation of the old corner definition or verifying
the alternative or creating a new one or alternatively
checked that the definition work in interesting cases
or to specify the cases where the current definition
or the alternative works.

I think that it may be so that:

8.
stable corner points S of a set has infinite corner rank ->
S is union of countably (finite or as many as integers) many
disconnected sets

I would be happy if that could be researched (it is enough to
be true hold also for some interesting cases). One reason is
that if the Mandelbrot-conjecture is true then I think that
there is a connection to the connectivity of Mandelbrot-set.
In a way it would be wasting time because it
known that Mandelbrot-set is connected but I think it may
be interesting. Even if the conjecture is false I would
like to know what do you or other people think of 8.
I have an impression that you Mr. Sheskey have the
knowledge to examine the special cases that may
make 8. false. Also I would like to know if you or
mathematicians find these ideas interesting, I have
some other ideas but they are not at least yet specific.



Also I correct a typo and specify what I mean (I wrote that it drops by
one):
S has finite corner rank <-> Hausdorff dimension of S is integer

S has finite corner rank and non-negative integer dimension <->
dimension of corners of S is a non-negative integer

(these should be true on interesting cases and as I wrote
differences appear when angled corners and non-angled corners
are take in account and infinite corner rank)

If you find ideas interesting I will collect to them to one place
for easy reference.


I think that it would be also interesting to consider the converse,
meaning what dimension a set has when the corners are known. It
may be true that the dimension increases by an integer. There
may be an interesting connection between derivation and integration
because the stripes around the Mandelbrot-set graphs of polynomials.


We Pretty

.

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