(08/07/2022, 02:50 PM)bo198214 Wrote: You (and Sheldon) are the ones with the numerical equipment of tinkering with the theta function, isn't it?
Don't put me on Sheldon's level, I am nowhere near Sheldon's level, lol. Especially when it comes to the crescent iteration. I am just indebted to his language when describing the beta method. I think it should be possible though, perhaps by beta iterating and letting the period be infinite.
Any function:
\[
\theta(z)\phi(z)\\
\]
Wouldn't be periodic, so it would be similar to "stretching the period" of the beta iteration I described for this LFT problem. This might be a good place to check if we can do "a crescent kind of iteration" with fibonacci, but using the "beta method". I will have to look into this. This seems really interesting. I mean, I always figured it possible, never thought I'd be able to hammer it down. And additionally was never that interested. This iterated function comparison has got me far more interested though.
What if this follows for more advanced recursions; what if iterations of functions \(p : \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}\), produces a correspondence to more general difference equations than Fibonacci. (Where I come from we make a change of variables to Fibonacci so that all linear difference equations are based on the operator \(\Delta = a_n - a_{n-1}\)). I know you explained a similar phenomenon for first order difference equations. But what if we increase the degree of \(p\) as a rational function--would it have a similar kind of correspondence? Wow, sounds super interesting. If anything, bo. You've given me A LOT to think about.
But technically, I have created a theta mapping which changes the period for this fibonacci iteration. It should be real valued. But, it'll probably have singularities on the real line, or about the real line. The trouble would be moving those singularities away from the real line, excepting a glaringly obvious pole somewhere on the real line (to account for the \(z=-1\) problem in the initial function). I think these beta solutions are more at home to mappings of \(\widehat{\mathbb{C}} \to \widehat{\mathbb{C}}\) ). So it may be real valued but less than desired. Again, I'm asking for an ENTIRE \(\theta(z)\), which makes this much more difficult. Only a crescent kind of iteration, which could or could not be constructed by "stretching the period" of the iteration.
This sounds like a good toy model though. I really like your use of LFTs, bo. This has given me something more "hold in your hands" to experiment with. I don't mean toy model offensively either, I hope you know that
