Im not sure how these relate but 3 ideas cross my mind.
( for every petal seperately : )
1)
the "realifier" mentioned by at minimum yourself and Bo.
2)
how about the 1-periodic theta functions present in the super functions ?
3)
vector calculus to " see " what is really going on and even perhaps solve it. Together with series expansions that might make more sense.
I mentioned fourier and L-series in the past.
Mittag-Leffler perhaps.
It seems taylor might not be the best here.
I assume you considered fractional calculus too. And the julia equation.
***
Also i had the idea
f^[t](x) = f_1(t) x + f_2(t) x^2 + f_3(t) x^3 + ...
Understanding these 3 functions seems key to me.
If those f_n agree with one of the iterates computed by the carleman matrix methods ( picking the correct roots ) then we are close to having the semi-group isom.
Notice f_1(t) also has the semi-group iso for both f_1(t) = 1 and for f_1(t) = (-1)^t.
In fact around a fixpoint f_1 defines bijectively a fitting theta function.
However the issues are not resolved ; radius 0 , divergeance , bad series expansions ? etc
But the truncated taylor function f_1(t) x + f_2(t) x^2 + f_3(t) x^3 should work well in the limit formula's I assume.
further ideas start to resemble the ideas around exp(x) - 1 ....
Ironically I think the branching thing is resolved in a way similar to your own ideas of that " realifier " picking branches... if we got the correct solution ofcourse.
I want to point out that dynamics on a plane is alot like liquid flow.
Not sure if that is of any help.
just my 50 cent.
regards
tommy1729
( for every petal seperately : )
1)
the "realifier" mentioned by at minimum yourself and Bo.
2)
how about the 1-periodic theta functions present in the super functions ?
3)
vector calculus to " see " what is really going on and even perhaps solve it. Together with series expansions that might make more sense.
I mentioned fourier and L-series in the past.
Mittag-Leffler perhaps.
It seems taylor might not be the best here.
I assume you considered fractional calculus too. And the julia equation.
***
Also i had the idea
f^[t](x) = f_1(t) x + f_2(t) x^2 + f_3(t) x^3 + ...
Understanding these 3 functions seems key to me.
If those f_n agree with one of the iterates computed by the carleman matrix methods ( picking the correct roots ) then we are close to having the semi-group isom.
Notice f_1(t) also has the semi-group iso for both f_1(t) = 1 and for f_1(t) = (-1)^t.
In fact around a fixpoint f_1 defines bijectively a fitting theta function.
However the issues are not resolved ; radius 0 , divergeance , bad series expansions ? etc
But the truncated taylor function f_1(t) x + f_2(t) x^2 + f_3(t) x^3 should work well in the limit formula's I assume.
further ideas start to resemble the ideas around exp(x) - 1 ....
Ironically I think the branching thing is resolved in a way similar to your own ideas of that " realifier " picking branches... if we got the correct solution ofcourse.
I want to point out that dynamics on a plane is alot like liquid flow.
Not sure if that is of any help.
just my 50 cent.
regards
tommy1729