Hmm, seems to make things better understandable. Here is the curve of f(x), but with focus on the aspect of also representing a trajectory of iterates. I made segments of iteration (each segment has range h modulo 1) visible by coloring: segments of equal color map consecutively to the next under iteration.
The "critical" area near the second fixpoint (read) is an area of oscillation, part of the green segment is overlaid by the red segment.
Interestingly the curve f(x) does not spiral like f°0.5(x), or let's say: it's a degenerate spiral, or it may be expressed as limit case of spiral without y-radius (or so...)
The light-blue segment of the f(x)-curve cannot be reached from any segment on the right; the iteration-"source" is imaginary. This answers the also the continuation of the f°0.5(x)-curve: it must (re-)appear in the complex plane.
Hmm...
I'm tending to following consideration:
* assume a function f(x)
* define one initial point (x0,f(x0))
* define the set of iterates -> discrete set of points, possibly accumulating at some fixpoints
* use binomial-theorem (our binomial-formula for iterates) to compose points to get continuous set of points.
The we have one segment of a curve within a certain interval (here the range between the two fixpoints -0.5<=x<=(1+sqrt(3))/2) , so to say: "one segment came into existence"
* then - how to express the generation of the rest of the curve for f(x) (the lightblue part in the plot) in this terms ?
Concerning the half-iterate:
remember - the current computation is based only on one special method to define a half-iterate.
* Can there be another approach to such interpolation, providing a non-spiraling curve for f°0.5(x) ?
* Or, let it be spiralling, but which crosses the y-axis exactly at (x,y) =(0,fp_y), where fp_y is the value of the fixpoint - this would be only a very small correction, but would look much more elegant...?
Gottfried
The "critical" area near the second fixpoint (read) is an area of oscillation, part of the green segment is overlaid by the red segment.
Interestingly the curve f(x) does not spiral like f°0.5(x), or let's say: it's a degenerate spiral, or it may be expressed as limit case of spiral without y-radius (or so...)
The light-blue segment of the f(x)-curve cannot be reached from any segment on the right; the iteration-"source" is imaginary. This answers the also the continuation of the f°0.5(x)-curve: it must (re-)appear in the complex plane.
Hmm...
I'm tending to following consideration:
* assume a function f(x)
* define one initial point (x0,f(x0))
* define the set of iterates -> discrete set of points, possibly accumulating at some fixpoints
* use binomial-theorem (our binomial-formula for iterates) to compose points to get continuous set of points.
The we have one segment of a curve within a certain interval (here the range between the two fixpoints -0.5<=x<=(1+sqrt(3))/2) , so to say: "one segment came into existence"
* then - how to express the generation of the rest of the curve for f(x) (the lightblue part in the plot) in this terms ?
Concerning the half-iterate:
remember - the current computation is based only on one special method to define a half-iterate.
* Can there be another approach to such interpolation, providing a non-spiraling curve for f°0.5(x) ?
* Or, let it be spiralling, but which crosses the y-axis exactly at (x,y) =(0,fp_y), where fp_y is the value of the fixpoint - this would be only a very small correction, but would look much more elegant...?
Gottfried
Gottfried Helms, Kassel