For the inspiration for this thread, see this thread
@sheldonison
I've tried to understand your Pari/GP scripts, but I think my fundamental issue is with the Kneser construction / Riemann mapping thing. So here is my understanding so far. Whatever the Kneser construction is, it seems to produce results consistent with regular iteration. Regular iteration produces power series with complex coefficients for \( b > \eta \), because the fixed points are complex for these bases. That makes sense to me. Regular iteration produces power series with real coefficients for \( \exp(-e) \le b \le \eta \), because the fixed points are real for these bases, or for all bases in the closed set of the region of convergence of the infinitely iterated exponential. I would also expect that tetration for "period 3" bases (approximately all complex bases with negative real part) to produce power series with complex coefficients, not only because the fixed points are complex, but also because of homotopy considerations, or that the orbits of 3 points that converge to a 3 cycle would require pushing any "lines" into a "round thing" (not sure if that's rigorous, but it makes sense to me).
So in this context, the Riemann mapping step is a method to find a function that somehow maps power series with complex coefficients to power series with real coefficients. The value of such a construction is that it allows us to compare regular iteration and intuitive iteration for \( b > \eta \). But there are too many unknowns for me: what are the properties of this Riemann mapping? how do we find it? what is the result? is it analytic? wouldn't this just be equivalent to
\( f^{-1}(x) = \text{tet}_b^{Reg}(\text{slog}_b^{Int}(x)) \)
and if this is the method for calculating the Riemann mapping, then we can't expect to learn anything about the two methods of iteration. Perhaps I should revisit this when I'm less confused.
@sheldonison
(01/13/2016, 01:37 PM)sheldonison Wrote: \( \text{tet}_b(z) =\;^z b= \exp_b^z(0) \)just as a point of order, I think you meant 1 instead of 0.
I've tried to understand your Pari/GP scripts, but I think my fundamental issue is with the Kneser construction / Riemann mapping thing. So here is my understanding so far. Whatever the Kneser construction is, it seems to produce results consistent with regular iteration. Regular iteration produces power series with complex coefficients for \( b > \eta \), because the fixed points are complex for these bases. That makes sense to me. Regular iteration produces power series with real coefficients for \( \exp(-e) \le b \le \eta \), because the fixed points are real for these bases, or for all bases in the closed set of the region of convergence of the infinitely iterated exponential. I would also expect that tetration for "period 3" bases (approximately all complex bases with negative real part) to produce power series with complex coefficients, not only because the fixed points are complex, but also because of homotopy considerations, or that the orbits of 3 points that converge to a 3 cycle would require pushing any "lines" into a "round thing" (not sure if that's rigorous, but it makes sense to me).
So in this context, the Riemann mapping step is a method to find a function that somehow maps power series with complex coefficients to power series with real coefficients. The value of such a construction is that it allows us to compare regular iteration and intuitive iteration for \( b > \eta \). But there are too many unknowns for me: what are the properties of this Riemann mapping? how do we find it? what is the result? is it analytic? wouldn't this just be equivalent to
\( f^{-1}(x) = \text{tet}_b^{Reg}(\text{slog}_b^{Int}(x)) \)
and if this is the method for calculating the Riemann mapping, then we can't expect to learn anything about the two methods of iteration. Perhaps I should revisit this when I'm less confused.