Real Multivalued/Parametrized Iteration Groups
#1
I was shock puzzled when Leo.M mentioned the possibility that a superfunction might not result
from or not have an iteration group.
This question arose from actual real valued superfunctions of \(b^z\) with b around eta minor.
There the fixed point derivative is negative and so would imaginably not allow for e.g. a half iterate.
However the given superfunctions are valid superfunctions S, satisfying
$$ S(x+1) = f(S(x)) $$
however they show oscillatory behaviour are hence not injective as we are used to from positive fixed point multipliers/derivatives.
In those cases you can just define
$$ f^{\circ t}(x) = S(t+S^{-1}(x)) $$
and you have an iteration (semi) group particularly satisfying
$$ f^{\circ s+t }(x) = f^{\circ s}(f^{\circ t}(x)),\quad f^{\circ 1} (x)=f(x) $$

So how can this be done in case S in not invertible?! Using Multivalued functions!

As a research object I started with the edge case \(f(x)=-x\) (multiplier -1).
This case is so edge that it often is not even called parabolic fixed point, because \(f(f(x))=x\).
However it serves our research purpose.
One can easily give the regular iteration group:
$$ f^{\mathfrak{R} t}(z) = (-1)^t z = e^{\pi i t}z $$
which is complex valued, the superfunction would be \(S^{\mathfrak{R}}(z)=e^{\pi i z}\).

Now we know from the real valued Fibonacci extension that this strange "realifyer" is used as a real replacement for \((-c)^t\)
$$ \rho(t) = \cos(\pi t) c^t $$
It satisfies \(\rho(t+1)=(-c)\rho(t)\) and \(\rho(1)=(-c)^1\) but is completely real valued.

And typically it can be used in conjuction with the inverse Schröder function, or the regular Superfunction to get a real valued solution, when we replace \((-c)^t \) with it:

So in our case we can define a superfunction for \(f(x)=-x\) by:
$$S(x)=\cos(\pi x) $$
I started with this edge case c=1 because I can easily invert this function, while it much more difficult when \(c\neq 1\).

So our iteration semigroup can be defined as
$$f^{\circ t}(x)= \cos\left(\pi \left(t\pm\frac{\arccos(x)}{\pi}\right)\right) = \cos(\pi t \pm \arccos(x)) $$
we need to choose the right branch of \(\arccos\).

       
Visually this gives an Aha! We have a circle as half iterate and and an ellipse as 1/3 iterate.
But we can check (pink line) that it indeed are proper iterates, as long as we choose the right branch,
though it does not even have the fixed point a 0 - very edge case.
And these are ellipses in the geometrical sense, the real thing - not such something that looks like it.
You can convince yourself when expanding \( \cos(\pi t \pm \arccos(x))\).



So this gives already a rich result for such a simple function, but it becomes even better for \(c\neq 1\).

       
Here instead of cumbersomely trying to compute inverse we just use a parametrized version,
which is quite simple to obtain an iteration group from a given Superfunction S:
$$ (x,y) = (S(s),S(s+t)) $$
that's it already.

This time the iterates have the same fixed point, as the original function, but - as they are not regular at the fixed point - they behave wildly around the fixed point (not too wild though).

So, conclusion:
Oscillating superfunctions are still associated with continuous iteration (semi)groups, only that they are multivalued now and one has to be careful with choosing the right branch (which is that way with most complex functions anyways).

   

Later I want to transfer that to our Fibonacci LFT - so stay tuned Wink
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#2
but those iterations probably fail hard on the complex plane ??

that is for complex x and/or s and/or t.


regards

tommy1729
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#3
This was the original comments by myself and MphLee when Leo first came on the scene. This is all Riemann surface territory. Riemann surface stuff gets wild when you talk about iterations. These are well known constructions; but it classifies itself under "dynamics on a Riemann surface". It behaves very similar to the normal dynamics we have; which is why I keep mentioning Milnor. We either have \(\mathbb{C}\) (Euclidean) or \(\mathbb{D}\) (A simply connected domain) or \(\widehat{\mathbb{C}}\) (The Riemann Surface). These are just three Riemann surfaces that we use as our base, especially with most of the work done here.

Choosing an arbitrary Riemann surface \(S\)--which can be as wild as possible, and taking iterates of \(f^{\circ n} : S \to S\)--we get everything Leo is talking about. The trouble is, talking about it as multivalued functions won't give you all the juice that Riemann surfaces will give you. It's like drinking orange juice compared to eating oranges. Whether we process our oranges first (Project the Riemann surface into the space of multivalued functions), or we just eat oranges (Prove everything with Riemann surfaces, that Milnor and most of complex dynamics set up).

If you want to go down the Riemann surface route though. We should refer to this more broadly. Where the preimage of the \(\beta\) tetration of \(\eta^-\) is Riemann surface, and constructing an action from this riemann surface to \(\mathbb{C}\) creates a multivalued semi group at \(\eta^-\).

This becomes:

$$
f^{\circ t}(z) = F\left(t+\mu(\mathcal{F}(z))\right)\\
$$

Where \(F\) is the beta iteration with period \(2 \pi i/\lambda\), and \(\mu : S \to \mathbb{C}\) and \(\mathcal{F}(z) = \{y \in \mathbb{C}\,|\, F(z) = y\}\). Then we just note that \(\{\forall z \mathcal{F}(z)\} = S\); and we are just choosing a projection schema (a multivalued function).



Let's go down the rabbit hole, bo! I loved Leo's original thesis. And I loved the idea. The trouble I had, was that it was the image, and not the Riemann surface preimage.
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#4
(08/17/2022, 02:58 AM)JmsNxn Wrote: This was the original comments by myself and MphLee when Leo first came on the scene. This is all Riemann surface territory. Riemann surface stuff gets wild when you talk about iterations. These are well known constructions; but it classifies itself under "dynamics on a Riemann surface". It behaves very similar to the normal dynamics we have; which is why I keep mentioning Milnor. We either have \(\mathbb{C}\) (Euclidean) or \(\mathbb{D}\) (A simply connected domain) or \(\widehat{\mathbb{C}}\) (The Riemann Surface). These are just three Riemann surfaces that we use as our base, especially with most of the work done here.

Choosing an arbitrary Riemann surface \(S\)--which can be as wild as possible, and taking iterates of \(f^{\circ n} : S \to S\)--we get everything Leo is talking about. The trouble is, talking about it as multivalued functions won't give you all the juice that Riemann surfaces will give you. It's like drinking orange juice compared to eating oranges. Whether we process our oranges first (Project the Riemann surface into the space of multivalued functions), or we just eat oranges (Prove everything with Riemann surfaces, that Milnor and most of complex dynamics set up).

If you want to go down the Riemann surface route though. We should refer to this more broadly. Where the preimage of the \(\beta\) tetration of \(\eta^-\) is Riemann surface, and constructing an action from this riemann surface to \(\mathbb{C}\) creates a multivalued semi group at \(\eta^-\).

This becomes:

$$
f^{\circ t}(z) = F\left(t+\mu(\mathcal{F}(z))\right)\\
$$

Where \(F\) is the beta iteration with period \(2 \pi i/\lambda\), and \(\mu : S \to \mathbb{C}\) and \(\mathcal{F}(z) = \{y \in \mathbb{C}\,|\, F(z) = y\}\). Then we just note that \(\{\forall z \mathcal{F}(z)\} = S\); and we are just choosing a projection schema (a multivalued function).



Let's go down the rabbit hole, bo! I loved Leo's original thesis. And I loved the idea. The trouble I had, was that it was the image, and not the Riemann surface preimage.

let me get this straith

$\mathcal{F}$ send you to the right complex value and mu sends you to the right branch ?

regards

tommy1729
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#5
(08/15/2022, 11:49 PM)tommy1729 Wrote: but those iterations probably fail hard on the complex plane ??

that is for complex x and/or s and/or t.


regards

tommy1729

right ?
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#6
Honestly, no idea Tommy. My knowledge of Riemann surfaces is largely cursory, and more so I learnt pieces of it by way of the link between Elliptic curves and Algebraic curves. In theory, there should be a lift (I believe that's the word), such that we can write:

$$
f^{\circ t} (z) : S \to S\\
$$

Where we are sort of sending the preimages to themselves. Then we can only care about the Riemann surface itself; and in proving things about the Riemann surface we prove things about the iteration on the complex plane--and figure out how and where it is holomorphic if it could be.


For example if \(S = \{ y | y = F(z)\}\) then we define a coordinate \(s_0 \in S\) such that \(f(s_0)\) is the value of \(y\) for \(F(z+1)\). And now we are acting on the preimage. Then we perform iterations as such.

I haven't done much Riemann surface stuff in a while, but this is kind of what you are doing. This isn't very different from how you define the Riemann surface of \(\log\). It's just \(\{y | y = e^z\}\).

Now we are talking about Dynamics on a Riemann surface; and from there, yes, you choose a projection of this riemann surface into the complex plane (you choose your branch of \(\log\)).
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#7
(08/17/2022, 02:58 AM)JmsNxn Wrote: If you want to go down the Riemann surface route though. We should refer to this more broadly.
(08/17/2022, 02:58 AM)JmsNxn Wrote: Let's go down the rabbit hole, bo! I loved Leo's original thesis. And I loved the idea. The trouble I had, was that it was the image, and not the Riemann surface preimage.

No, I totally don't want to go that rabbit hole! This would be a lot of theory to digest, with possibly no effect on the questions we have - rather it would pose a lot of new questions! Remember, I am beyond my limit already ...

And I found already what I wanted. Its not just that you can say: well we doing everything multivalued (like when you draw a picture of arctan with stacked copies along y) and somehow some choice of branch will work - but it is rather the idea of (analytic) continuation that solves the puzzle - the graph is *one* continuous line, one does not need to care which branch cut gets glued to which other branch cut.

But what exactly you mean by Leo's original thesis?
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#8
(08/18/2022, 06:56 AM)bo198214 Wrote: But what exactly you mean by Leo's original thesis?

Oh, I apologize here. Leo made a bunch of introductory posts about a year or so ago. And it's very similar to this construction. And Mphlee and I were talking to him about it, and asking him questions. So I'm a big fan of the idea when he first provided it. I don't mean thesis, thesis. I Just mean the constructions he's developed when referring to multivalued iterations. And I'm just referring to it as a thesis, as though this is an idea he's originated and "defending like a thesis".

Again, Bo. I apologize, it's the casual nature of my language. I don't mean thesis, I mean his introduction and discussion from a year or so ago.
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#9
(08/18/2022, 07:30 AM)JmsNxn Wrote:
(08/18/2022, 06:56 AM)bo198214 Wrote: But what exactly you mean by Leo's original thesis?

Oh, I apologize here. Leo made a bunch of introductory posts about a year or so ago. And it's very similar to this construction. And Mphlee and I were talking to him about it, and asking him questions. So I'm a big fan of the idea when he first provided it. I don't mean thesis, thesis. I Just mean the constructions he's developed when referring to multivalued iterations. And I'm just referring to it as a thesis, as though this is an idea he's originated and "defending like a thesis".

Again, Bo. I apologize, it's the casual nature of my language. I don't mean thesis, I mean his introduction and discussion from a year or so ago.

There is totally no need to apologize for that I asked a question!
(And in a lot of other posts you made there was also no reason at all to apologize ... there is no fault on you)
Also I didn't read it in the sense of master thesis or phd thesis.
Just wanted to know what it is!
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#10
Here you go!

https://math.eretrandre.org/tetrationfor...p?tid=1318

Where Leo. W as a user makes their introduction and draws out a lot of cool shit, and we bicker a bit about Riemann surfaces, lol.

Then,

https://math.eretrandre.org/tetrationfor...p?tid=1539

Where we enter a lot of observations by Mphlee, which are relevant when we concern ourselves with the group property \(f \circ g = g \circ f\) and the admission of an exponential \(f \circ g(t) = g(t+1)\).

This is where Leo makes his introduction, and I mean that as an "introduction to a good paper".
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