08/19/2022, 07:09 PM
To be honest this scenario is not "real life" to me. So even if you could prove it, it's not applicable to most cases (e.g. one of the fixed points is repelling and we don't talk about entire functions.)
The typical scenario is that we have some real analytic function with two adjacent fixed points (one attracting one repelling) with multiplier > 0, we know the regular iteration at each point asking whether both iterations are equal, or equivalently one iteration can be continued beyond the other fixed point. Similar (because the limit case of both hyperbolic fixed points get united) scenario: we have one parabolic fixed point with multiplier 1 and want to know whether the regular iteration is analytic there.
So the answer that was given by Karlin& McGregor (hyperbolic case) is: If the function is analytic on C except a closed countable set of isolated singularities, then it can not have such two fixed points except for LFTs. Similar (and earlier) result of Baker (parabolic case): If the function is meromorphic it can not have such fixed point except for LFTs. (And I even found a result of Liverpool that extends the set of functions similar to the one in the hyperbolic case: meromorphic single valued function with the exception of at most countable isolated singularities - I attach the paper to this post. It's theorem 3).
These statements still leave a lot of room for other kinds of functions where two fixed points can have the same regular iteration.
Also not nice in your assumptions is that you hide an aspect of the function that has nothing to do with continuous iteration, but comes already from integer iteration: Whether a singularity would come arbitrarily close to one fixed point. Your assumption looks already as if it only allows entire functions (in the case of repelling fixed point) - without even utilizing continuous iteration. Also "local iteration" seems not a very suitable term anymore, because you don't talk about a vicinity of a point, but a full fledged domain.
Your assumptions would imho be more reasonable if you wanted to take the functional limit of infinite iteration. There I would understand these assumptions.
I mean great if you intuitively know these properties from your extended study of iteration (theory), but I don't - totally don't - see it. So I would need a very small stepped proof. On the other hand you should rather invest your efforts to more fruitful enterprises than proving this statement!
The typical scenario is that we have some real analytic function with two adjacent fixed points (one attracting one repelling) with multiplier > 0, we know the regular iteration at each point asking whether both iterations are equal, or equivalently one iteration can be continued beyond the other fixed point. Similar (because the limit case of both hyperbolic fixed points get united) scenario: we have one parabolic fixed point with multiplier 1 and want to know whether the regular iteration is analytic there.
So the answer that was given by Karlin& McGregor (hyperbolic case) is: If the function is analytic on C except a closed countable set of isolated singularities, then it can not have such two fixed points except for LFTs. Similar (and earlier) result of Baker (parabolic case): If the function is meromorphic it can not have such fixed point except for LFTs. (And I even found a result of Liverpool that extends the set of functions similar to the one in the hyperbolic case: meromorphic single valued function with the exception of at most countable isolated singularities - I attach the paper to this post. It's theorem 3).
These statements still leave a lot of room for other kinds of functions where two fixed points can have the same regular iteration.
Also not nice in your assumptions is that you hide an aspect of the function that has nothing to do with continuous iteration, but comes already from integer iteration: Whether a singularity would come arbitrarily close to one fixed point. Your assumption looks already as if it only allows entire functions (in the case of repelling fixed point) - without even utilizing continuous iteration. Also "local iteration" seems not a very suitable term anymore, because you don't talk about a vicinity of a point, but a full fledged domain.
Your assumptions would imho be more reasonable if you wanted to take the functional limit of infinite iteration. There I would understand these assumptions.
I mean great if you intuitively know these properties from your extended study of iteration (theory), but I don't - totally don't - see it. So I would need a very small stepped proof. On the other hand you should rather invest your efforts to more fruitful enterprises than proving this statement!
