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+--- Thread: fractals and superfunctions for f(x,y) ? (/showthread.php?tid=1639)
fractals and superfunctions for f(x,y) ? - tommy1729 - 09/15/2022
Consider the function f(x,y) being a mapping from R^2 to R^2 such that
f(x,y) = ( taylor1(x,y) , taylor2(x,y) )
where taylor1 and taylor 2 do not satisfy the cauchy riemann equations , hence f(x,y) is not isomorphic to an analytic function f(z) with real x and im y.
f^[2](x,y) = ( taylor1(f(x,y)) , taylor2(f(x,y)) ) = ( taylor1( taylor1(x,y),taylor2(x,y) ) , taylor2( taylor1(x,y),taylor2(x,y) ) )
and in general for t >= 0
f^[0](x,y) = (x,y)
f^[t](x,y) = ( taylor1( f^[t-1](x,y) ) , taylor2( f^[t-1](x,y) ) )
(with some abuse notation sorry )
How about fractals and superfunctions for these f(x,y) ?
Maybe take taylors to be real polynomials to start.
many analogues must exist to ideas from complex dynamics.
Leo considered a rotation.
fixpoints might occur.
but also saddle points.
inversion might be troublesome : e.g. x^2 + y^2 = 1 has uncountable solutions.
In particular consider the case from a region A to region B by f(x,y) such that the mapping is injective.
btw i use taylor to avoid piecewise functions ( even c^oo can be piecewise ! ) which imo is too general.
So basically dynamics of mappings on a plane that are injective but not holomorphic or antiholomorphic.
it relates to dynamics , chaos and differential equations ofcourse.
but i want to know what you think and know.
the semi-group property is slightly desired
regards
tommy1729
RE: fractals and superfunctions for f(x,y) ? - bo198214 - 09/15/2022
(09/15/2022, 12:25 PM)tommy1729 Wrote: f^[2](x,y) = ( taylor1(taylor1(x,y)) , taylor2(taylor2(x,y)) )
I thought that would be:
f^[2](x,y) = ( taylor1(taylor1(x,y),taylor2(x,y)) , taylor2(taylor1(x,y),taylor2(x,y)) )
?
RE: fractals and superfunctions for f(x,y) ? - tommy1729 - 09/15/2022
(09/15/2022, 12:56 PM)bo198214 Wrote:(09/15/2022, 12:25 PM)tommy1729 Wrote: f^[2](x,y) = ( taylor1(taylor1(x,y)) , taylor2(taylor2(x,y)) )
I thought that would be:
f^[2](x,y) = ( taylor1(taylor1(x,y),taylor2(x,y)) , taylor2(taylor1(x,y),taylor2(x,y)) )
?
yes ofcourse you are right. thanks! I was in a hurry.
I corrected it with slight abuse notation ...
regards
tommy1729
RE: fractals and superfunctions for f(x,y) ? - tommy1729 - 09/15/2022
Ofcourse one thinks immediately about
1) the case isolated attracting fixpoint ;
** the analogue of koenigs **
jacobian matrix for linear approximation near the fixpoint, then using Hartman-Grobman theorem/theory to justify the analogue construction.
2) the henon map
3) Kolmogorov-Arnold-Moser theorem
https://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman_theorem
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem#:~:text=The%20KAM%20theorem%20states%20that,is%20continuous%20in%20the%20perturbation.
4) bifurcation (theory)
5) the trivial case f(x,y) = ( g(x) , y ) basically reducing to a 1 variable case.
and probably the ideas in this paper and the analogues similar to jacobian ;
( added file )
6) what about the case when there is ( locally ) no fixpoint ?
***
This also related to an idea of mine that the group addition isomorphism property could be tested here and shown to be a uniqueness criterion.
in the case of 1) this is probably not so hard.
In the case of 6) probably harder.
I had the idea of showing that the group addition isomo implies the cauchy riemann eq for the superfunction when the considered base function is analytic , and thus the super also being analytic but that is a conjecture in the analytic domain so kinda off topic , yet related.
***
***
I also think the gaussian method can be extended in this way in many cases.
***
***
remark :
*Under the group addition isomo *
for clarity I consider ( locally when we have injection ) f^[t + t_2 i ] with the imag part as perpendicular between two real t paths of different starting values.
so f^[t_2 i](x_a,y_a) = f^[t](x_b,y_b) = f^[t + t_2 i](x,y)
where the paths of real iterates of (x_a,y_a) and (x_b,y_b) do not intersect in the injective domain.
Since imag numbers are not existant in this " universe " maybe writing
f^[t x + t_2 y] is more logical , yet essentially the same.
***
regards
tommy1729
RE: fractals and superfunctions for f(x,y) ? - tommy1729 - 09/16/2022
related :
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1641