09/15/2022, 11:36 PM
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 ?
***
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
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 ?
***
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