the inconsistency depending on fixpoint-selection

[Gottfried]

However your second transformation
\( x' = -4 x +1 \), \( x'' = -\frac{1}{4} x + \frac{1}{4} \), \( g_1(x)=f(x'')' \)
has as translation \( \tau(x)=x+\frac{1}{4} \) the same translation as in your previous transformation!
Only \( \mu \) changes to \( \mu(x)=-\frac{1}{4}x \).

So this second transformation moves also the *greater* fixed point to 0. And we have seen that we can dismiss any multiplicative conjugation \( \mu^{-1}\circ f\circ \mu \) with respect to regular iteration. So it has of course the same result (fractional iteration of \( f \)) as your first transformation.

Yepp - I just found the position of the error, grmmpff. It is in

Code:

´
using e1=1/4                                e2 = -1/4  
      d1=1/4 / e1 =1                        d2 = 1/4 / e2 = -1

x' =  4x - 1                               x´ = -4x + 1
x" =  (x+1)/4                              x´´= (x-1)/-4

Although I had already that not e or d, but e*d = fixpoint, in the computation of d2, I used the same (1/4) as for d1. For clarity, I should have written a symbol for the selected fixpoint:

Code:

´
involving fixpoint p=1/4                    fixpoint q=-1/4
using e1=1/4                                e2 = -1/4  
      d1=p / e1 =1                          d2 = q / e2 = 1

x' =  4x - 1                               x´ = -4x - 1
x" =  (x+1)/4                              x´´= (x+1)/-4

Hope this corrects things, didn't check it again yet, but it seems that this reflects just the simple sign-changing in the matrices.
The previous G2-matrix and the subsequent considerations in that posting are on a false assumption...

Gottfried