Just a small result from some playing around in a break; I'll look at the confirmation for a wider range of parameters later.
This days I'm re-considering the differences of the fractional iterates of \( b^x -1 \) using the schröder-function by their carleman-matrices. At the moment I use b=2, comparing the results when I use the power series centered around fixpoint t0=0 versus that around fixpoint t1=1. The range of 0<x0<1 can be analyzed well using both power series and we get the known effect, that the iterates disagree at fractional heights (but not at integer heights), and that the differences form a sinusoidal curve.
If we compare the according fractional iterates which we get using the Newton-formula (the binomial composition of integer-height-iterates only) which does not use a fixpoint, then it seems, that the results approximate that of the powers series around zero. I think this means for more general bases (other than 2): around the attracting fixpoint, which is zero only if log(base)<1 but is the negative real fixpoint if log(base)>1.
The point x0, at which the most symmetric sinusoidal curve occurs seems to be x0~0.382160520000... (not rational!) which has a special property for base b=2 which I'll discuss in a later post.
Gottfried
This days I'm re-considering the differences of the fractional iterates of \( b^x -1 \) using the schröder-function by their carleman-matrices. At the moment I use b=2, comparing the results when I use the power series centered around fixpoint t0=0 versus that around fixpoint t1=1. The range of 0<x0<1 can be analyzed well using both power series and we get the known effect, that the iterates disagree at fractional heights (but not at integer heights), and that the differences form a sinusoidal curve.
If we compare the according fractional iterates which we get using the Newton-formula (the binomial composition of integer-height-iterates only) which does not use a fixpoint, then it seems, that the results approximate that of the powers series around zero. I think this means for more general bases (other than 2): around the attracting fixpoint, which is zero only if log(base)<1 but is the negative real fixpoint if log(base)>1.
The point x0, at which the most symmetric sinusoidal curve occurs seems to be x0~0.382160520000... (not rational!) which has a special property for base b=2 which I'll discuss in a later post.
Gottfried
Gottfried Helms, Kassel