I am not sure I get your problem correctly.
Take the function \( f: b^z \) as to be iterated, with, say \( b=sqrt(2) \) .
Assume one plane on a math-paper and look for easiness only the lines and their crossings of the coordinate-system of integer complex numbers \( z_0 \) .
Now take another paper, position it 10 cm above and for every point of the crossings (and ideally also of the lines) mark the values of \( z_1=b^{z_0} \). Then repeat it with a third plane, again 10 cm above, marking \( z_2=b ^{b^{z_0}} \) .
After that, try to connect the related points of the zero'th, the first and the second plane by a weak string, say a spaghetti or so. Surely except of the fixpoints in \( z_0 \) it shall be difficult to make a meaningful and smooth curve - and in principle it seems arbitrary, except at the fixpoints, where we simple stitch a straight stick through the iterates of the \( z_0 \) at the fixpoint.
Of course the spaghetti on the second level is then no more arbitrary but must be - point for point - be computed by one iteration. But the spaghatti in the first level follow that vertically orientated curve, where a fictive/imaginative plane of paper is at fractional heights and the fractional iterates would be the marks on the coordinate-papers at the "fractional (iteration) height".
I'd liked to construct some physical example, showing alternative paths upwards between the fixed basic planes, with matrial curves made by an 3-D-printer, but I've not yet started to initialize the required data.
But I think, that mind-model alone makes it possibly already sufficiently intuitive for you.
A somewhat better illustration is in my answer at MSE, see https://math.stackexchange.com/a/451755/1714
Take the function \( f: b^z \) as to be iterated, with, say \( b=sqrt(2) \) .
Assume one plane on a math-paper and look for easiness only the lines and their crossings of the coordinate-system of integer complex numbers \( z_0 \) .
Now take another paper, position it 10 cm above and for every point of the crossings (and ideally also of the lines) mark the values of \( z_1=b^{z_0} \). Then repeat it with a third plane, again 10 cm above, marking \( z_2=b ^{b^{z_0}} \) .
After that, try to connect the related points of the zero'th, the first and the second plane by a weak string, say a spaghetti or so. Surely except of the fixpoints in \( z_0 \) it shall be difficult to make a meaningful and smooth curve - and in principle it seems arbitrary, except at the fixpoints, where we simple stitch a straight stick through the iterates of the \( z_0 \) at the fixpoint.
Of course the spaghetti on the second level is then no more arbitrary but must be - point for point - be computed by one iteration. But the spaghatti in the first level follow that vertically orientated curve, where a fictive/imaginative plane of paper is at fractional heights and the fractional iterates would be the marks on the coordinate-papers at the "fractional (iteration) height".
I'd liked to construct some physical example, showing alternative paths upwards between the fixed basic planes, with matrial curves made by an 3-D-printer, but I've not yet started to initialize the required data.
But I think, that mind-model alone makes it possibly already sufficiently intuitive for you.
A somewhat better illustration is in my answer at MSE, see https://math.stackexchange.com/a/451755/1714
Gottfried Helms, Kassel
