fractals and superfunctions for f(x,y) ?

[tommy1729]

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

[bo198214]

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)) )

?

[tommy1729]

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

[tommy1729]

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...an_theorem

https://en.wikipedia.org/wiki/Kolmogorov...rturbation.

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 ?

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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.

 
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I also think the gaussian method can be extended in this way in many cases.

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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.


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regards

tommy1729

[tommy1729]

related :

https://math.eretrandre.org/tetrationfor...p?tid=1641