08/05/2022, 01:20 AM
Oooooooooo very excited by this.
I think I see how you can derive that the function \(q_t(s)\) is periodic in \(s\) too (I'm assuming for fixed \(t\), but the period depends on \(t\)). I wrote a bunch but I scrapped it because I had assumed something that was false, but it's looking like it should be inverse mellin transformable/differintegrable. And then you can use that since \(q_n(s)\) is constant (hence periodic (or else you wouldn't say that)--no rational function is periodic unless it's constant) to derive periodicity for each \(t\), but I'm not certain yet.
Also, I think this would only qualify as a local iteration if we allowed poles. Where then we are talking about functions on \(\widehat{\mathbb{C}}\). So a local iteration about two points independent of \(t\) will probably be possible, but once extended to its maximal domain will have poles. There's a reason \(\widehat{\mathbb{C}}\) is treated as its own beast in Milnor (he calls it spherical), it changes the game entirely. The euclidean case, is specific to transcendental entire functions; that have rules of their own. So I'm still not convinced that, in the euclidean case, you've provided a counter example. But still, you are widening my eyes as to what I've assumed
!
Also, I don't think you even need to put in as much work as you've done to find a counter example. Taking conjugations of \(\lambda^t z\) by linear fractional transformations, will probably already supply that. I'll need to work harder to carve out exactly what the theorem to be. Namely, that it should be only specific to Euclidean transformations (Transcendental entire functions).
I think I see how you can derive that the function \(q_t(s)\) is periodic in \(s\) too (I'm assuming for fixed \(t\), but the period depends on \(t\)). I wrote a bunch but I scrapped it because I had assumed something that was false, but it's looking like it should be inverse mellin transformable/differintegrable. And then you can use that since \(q_n(s)\) is constant (hence periodic (or else you wouldn't say that)--no rational function is periodic unless it's constant) to derive periodicity for each \(t\), but I'm not certain yet.
Also, I think this would only qualify as a local iteration if we allowed poles. Where then we are talking about functions on \(\widehat{\mathbb{C}}\). So a local iteration about two points independent of \(t\) will probably be possible, but once extended to its maximal domain will have poles. There's a reason \(\widehat{\mathbb{C}}\) is treated as its own beast in Milnor (he calls it spherical), it changes the game entirely. The euclidean case, is specific to transcendental entire functions; that have rules of their own. So I'm still not convinced that, in the euclidean case, you've provided a counter example. But still, you are widening my eyes as to what I've assumed
!Also, I don't think you even need to put in as much work as you've done to find a counter example. Taking conjugations of \(\lambda^t z\) by linear fractional transformations, will probably already supply that. I'll need to work harder to carve out exactly what the theorem to be. Namely, that it should be only specific to Euclidean transformations (Transcendental entire functions).