Hi,
I was just reviewing some properties of the dxp-function, the map: x-> b^x-1 .
For functions without constant terms it is widely common to accept the fractional iterate via the powerseries generated by fractional matrix-powers as "natural" embedding into the continuous flow - and to ignore issues of other fixpoints (besides the fixpoint zero). However, I looked at the powerseries, for instance 2^x-1, developed at the other fixpoint - and not surprisingly, the fractional iterates differ in the range for x, where the range of convergence of the different power series for their schroeder-functions overlap, as well as we've discussed this with x-> b^x (where b=e or b=sqrt(2) etc) for instance in the "bummer" thread.
So at this moment I became curious about the common acceptance of the "natural" embedding of iterations of 2^x-1 or exp(x)-1 or in general of functions having f(0)=0 and f'(0)<>0 in a flow based on their powerseries around zero.
Has someone come across that discussion in the context of the dxp? A discussion about the "wobbling" - the sinusoidal different solutions per unit-height-iteration-interval when centered around the different fixpoints? And an explicite justification for the common choice?
I was just reviewing some properties of the dxp-function, the map: x-> b^x-1 .
For functions without constant terms it is widely common to accept the fractional iterate via the powerseries generated by fractional matrix-powers as "natural" embedding into the continuous flow - and to ignore issues of other fixpoints (besides the fixpoint zero). However, I looked at the powerseries, for instance 2^x-1, developed at the other fixpoint - and not surprisingly, the fractional iterates differ in the range for x, where the range of convergence of the different power series for their schroeder-functions overlap, as well as we've discussed this with x-> b^x (where b=e or b=sqrt(2) etc) for instance in the "bummer" thread.
So at this moment I became curious about the common acceptance of the "natural" embedding of iterations of 2^x-1 or exp(x)-1 or in general of functions having f(0)=0 and f'(0)<>0 in a flow based on their powerseries around zero.
Has someone come across that discussion in the context of the dxp? A discussion about the "wobbling" - the sinusoidal different solutions per unit-height-iteration-interval when centered around the different fixpoints? And an explicite justification for the common choice?
Gottfried Helms, Kassel