04/30/2021, 05:32 AM
I'll take time to read over your answer more in depth later; but as I understand it, and as I'm boiling it down; it's a question of how the Kneser function chooses it's logarithms.
If we fix a logarithm, and we take a cycle of \( z_k \); it is not necessarily true that,
\(
\log(z_j) = z_{j-1}\\
\)
But, there exists a logarithm where this is true; it's some trailing multiple of \( 2\pi i \) away. However, the Kneser function knows to do this; such that the map \( \log \) will necessarily choose the right branch (or rather, the right sequence of branches). I presume by your Key; you mean a sequence of numbers \( (k_0,...,k_{n-1}) \) such that,
\(
\log_{k_j}(z_j) = z_{j-1}\\
\)
And,
\(
\log_{k_j}(z) = \log(z) + 2\pi i k_j\,\,\text{for the principal branch of }\log\\
\)
So I'm seeing this as a goal to describe what sequence of logarithms work. Correct me if I'm way off here; I'm just trying to understand.
Do cycles stay as cycles indefinitely in Kneser's iteration (I believe this can't happen as \( \Re(z) \to -\infty \) it causes \( \text{tet}_K(z) \to L,L* \); and this limit should be satisfied clearly.) So Kneser's "keys" if you will will eventually degrade and no longer work; but as you do the forward iteration we get the usual weird cluster of repelling cycles. And if we pull back from the forward iteration; we have to keep track of which logarithm, when.
Are you asking if we can use different keys to construct a varied Tetration? Or at least a different method of computation?
I'll read your treatise tomorrow; need to go to bed tonight. Thank you for your explanation though; I'll read it more carefully tomorrow.
Regards, James
If we fix a logarithm, and we take a cycle of \( z_k \); it is not necessarily true that,
\(
\log(z_j) = z_{j-1}\\
\)
But, there exists a logarithm where this is true; it's some trailing multiple of \( 2\pi i \) away. However, the Kneser function knows to do this; such that the map \( \log \) will necessarily choose the right branch (or rather, the right sequence of branches). I presume by your Key; you mean a sequence of numbers \( (k_0,...,k_{n-1}) \) such that,
\(
\log_{k_j}(z_j) = z_{j-1}\\
\)
And,
\(
\log_{k_j}(z) = \log(z) + 2\pi i k_j\,\,\text{for the principal branch of }\log\\
\)
So I'm seeing this as a goal to describe what sequence of logarithms work. Correct me if I'm way off here; I'm just trying to understand.
Do cycles stay as cycles indefinitely in Kneser's iteration (I believe this can't happen as \( \Re(z) \to -\infty \) it causes \( \text{tet}_K(z) \to L,L* \); and this limit should be satisfied clearly.) So Kneser's "keys" if you will will eventually degrade and no longer work; but as you do the forward iteration we get the usual weird cluster of repelling cycles. And if we pull back from the forward iteration; we have to keep track of which logarithm, when.
Are you asking if we can use different keys to construct a varied Tetration? Or at least a different method of computation?
I'll read your treatise tomorrow; need to go to bed tonight. Thank you for your explanation though; I'll read it more carefully tomorrow.
Regards, James
