Very interesting, Leo!
This is probably related to the fundamental group of the Riemann Surface of \( g \). Where you have a general Riemann surface of the iterates; and there exists 4 canonical projections to \( \mathbb{C} \); and we then create an algebra from them. Something like that--it may not be the fundamental group that you're describing; but it definitely borders it. Can't be certain, but it definitely looks like it.
This is again, though; why Riemann surfaces are so important. The algebra unveils in a general setting.
Very interesting, though. I like the rough work of it you did. Very cool!
Regards, James
You definitely got me thinking. I think I can write this a tad more general; but I'd prefer starting from \( f(z) = z - 1/z \) because it's got simpler features. Then generalize to how much more complex \( z+1/z \) is.
This is probably related to the fundamental group of the Riemann Surface of \( g \). Where you have a general Riemann surface of the iterates; and there exists 4 canonical projections to \( \mathbb{C} \); and we then create an algebra from them. Something like that--it may not be the fundamental group that you're describing; but it definitely borders it. Can't be certain, but it definitely looks like it.
This is again, though; why Riemann surfaces are so important. The algebra unveils in a general setting.
Very interesting, though. I like the rough work of it you did. Very cool!
Regards, James
You definitely got me thinking. I think I can write this a tad more general; but I'd prefer starting from \( f(z) = z - 1/z \) because it's got simpler features. Then generalize to how much more complex \( z+1/z \) is.