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This follows from the discussion held at: Jabotinsky IL and Nixon's program: a first categorical foundation, (May 10, 2021), Tetration Forum.
(05/12/2021, 11:10 PM)JmsNxn Wrote: Mphlee and I are discussing a manner of classifying conjugate classes of holomorphic functions. Which, naively, one would write[...]
\( [f,g]:=\{\chi:X\to Y:\chi\circ f=g\circ\chi\} \)
Here JmsNxn is touching an interesting terminological problem.
I don't know yet how to call those classes.
The problem
I'd not call them conjugacy classes. The compositional-Residual classes\( {\rm Rsd}(f,\zeta) \)that JmsNxn describes in his theory seems to undoubtedly deserve the term "conjugacy classes" (or subsets of conjugacy classes) because of compositional-Cauchy theorem. With that I mean that to deserve the name of conjugacy class a set has to have all of its elements pairwise conjugated by something. For example the set of even numbers \( [2]_2=[0]_2\in{\mathbb Z}/2{\mathbb Z} \) is an equivalence class of integers mod 2. All the even integers are related by being even.
Our sets [F,G] are hom sets (in some category), i.e. set of homorphisms that preserves some structure (that of the iteration). Calling them just hom-sets it's not enough informative. In the last months we called them "supefurfunctions" from F to G.
The case against for "superfunctions".
Perhaps a effective name that can resonate with this forum would be set of superfunctions from F to G but I fear Kouznetsov could claim the term for a more precise definition (@Mizugadro): [F,G] when F=S the successor, G holomorphic and when we are in the complex numbers. The things inside [F,G] remain a more general kind of objects.
Following the theory of dynamical system.
In the context of (eventually topological) dynamical systems, if we restrict to the subset of [F,G] invertible functions, we could call them "topological conjugations" between F and G. So [F,G] should be called the "clas of conjugations"? But... the things in [F,G] need not to be invertible, they are not, to be pedantic, conjugations in the true meaning of the word plus with the term "conjugation" it's meant in group theory an inner automorphism or the equivalence relation.
It helps me to picture them "as if they were conjugations." The positive side of this is that in monoid and semigroup theory (where invertibility is not required) they call conjugation the generalizations of group conjugacy.
The case for equivariant maps.
In the literature ( wikipedia+ the n-Lab) the term employed for morphism of monoid action is that of "equivariant map". If we see \( F:X\to X \) and \( G:Y\to Y \) not as a mere functions but as a T-flows extending the iteration of F and G to a monoid of time \( T \) then the equation
\( \chi (F(t,x))=G(t,\chi(x)) \)
that \( \chi:X\to Y \) satisfies is called equivariance. E.g.: morphisms of vector spaces (linear applications) satisfy equivariance+other things (\( F(\lambda{\bf v})=\lambda F({\bf v}) \)).
I conclude that a better term, if we look far for generalizations, should be: [F,G] is set(class) of \( \mathbb N \)-equivariant maps.
I'm open to critiques and opinions.
Regards
MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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Personally, I'm a fan of conjugate classes; but you're right; technically that's incorrect. I would usually always consider invertible (at least, locally) maps. So, in that frame work it's correct. I do like equivariant, but that seems a tad restrictive when we're just considering one variable. Superfunction already seems reserved. Change of base honestly sounds like the best one, if you could only make this a better term than just change of base. It also fits more at home with this forum, where it is an equivalent to what we typically call change of base.
I wonder if there is a term for conjugate classes in monoids? Or at least an equivalent for monoids?
I'll have to think about this. I think you're right, we need a good term for this.
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Thank you for the time James. I apologize for some grammar errors.
(05/14/2021, 03:36 PM)JmsNxn Wrote: Personally, I'm a fan of conjugate classes; but you're right; technically that's incorrect. I would usually always consider invertible (at least, locally) maps. So, in that frame work it's correct.
I'm sorry but here you lose me. In which local framework elements of [f,g] can be thought as part of a conjugacy class in the literal meaning of the term?
I call two elements f,g related by conjugation (conjugated) if exists, at least, a third object x s.t. xf=gx. In the case of \( {\rm Rsd}(f,\zeta) \) we have that each pair is conjugated by a path, or more then one, (by your compositional-Cauchy thm (CCT)). Now, if there exists more objects x,y,z that conjugate f and g, i.e.
xf=gx, yf=gy adn zf=gz
we can say that x,y,z are three different ways in which f and g are in relation (conjugated). We can see x,y,z as modes of being conjugated of f and g. Using the geometric intuition of curves and your CCT we can see x,y,z as oriented paths joining f to g.
If we want to retain the conceptual connection with conjugacy, which is fine to me, I find appropriate to call [f,g] the set of modes of f to being conjugated to g.
Some alternatives: - [f,g] is the set of oriented paths from f to g
- [f,g] is the set of transformations of f into g
- [f,g] is the set of f-iterations of g or of f-superfunctions of g
- [f,g] is the set of g-logarithms of f or of g-abel functions of f
Sadly, the relation of being conjugated, i.e. conjugation, is an equivalence relation only if we are in a group. Only in that case it defines a partition into equivalence classes of the ambient.
Are the sets [f,g] equivalence classes of some equivalence relation?
Nope, because they are never disjoint as far as gorups, monoids and semigroups are concerned. If we collect all the [f,g] they do not form a partition of the total set of functions (to prove this just consider [id,id] everything commutes with the identity).
It is an open problem if elements of [f,g] are conjugated and how to classify them. Let x,y such that xf=gx and yf=gy... When is there an object s such that sx=ys?
Quote: I do like equivariant, but that seems a tad restrictive when we're just considering one variable.
What do you mean?
Quote:Change of base honestly sounds like the best one, if you could only make this a better term than just change of base. It also fits more at home with this forum, where it is an equivalent to what we typically call change of base.
I'm not sure. Change of coordinates system?
I was thinking of change of time: because an element x conjugating f to g changes the frame of reference in some sense... (that's the case of linear algebra) but since we are talking about two iterations maybe we could force the analogy by claiming that x is translating the time of f to the time of g.
This can be confusing if we think of changing time as changing our monoid of time, e.g. when we extend from N to Z, or from integer iteration (Z) to complex iteration ©.
But I've a proof (that has to do with rank=1 functions) that the change of time monoid is related to some extent to the idea that \( \chi \in [f,g] \) is really changing the time from f to the time g.
Quote:I wonder if there is a term for conjugate classes in monoids? Or at least an equivalent for monoids?
Yes. As I was saying, the relation of conjugacy, as I defined it, is an equivalence only for groups. In monoids and semigroups the possible definitions of conjugacy become dependent on orientation and on other technical difficulties. In other words the concept fragments in a couple of variants. In "the literature", as far as I know and remember that I'm just a nobody that didn't get past the first semester of university, they still call them conjugacy classes. Usually, all the different possible definitions are equivalent when we restrict ourselves to groups.
The two most striking ones, that hold universally in semigroups and monoids (i.e semigroups with identity element), are the relations of isomorphism and of connectedness. These two relations define classes that are called isomorphism classes and, respectively, connected components. (note: this is not fringe math but very well know. See Tom Leinster on MO.)
As a side note: I'd love to hear opinions of historical heavyweights of the forum like Trappmann, Robbins, Sheldonison, Tommy, Gottfried and Mick
MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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(05/14/2021, 06:42 PM)MphLee Wrote: Thank you for the time James. I apologize for some grammar errors.
(05/14/2021, 03:36 PM)JmsNxn Wrote: Personally, I'm a fan of conjugate classes; but you're right; technically that's incorrect. I would usually always consider invertible (at least, locally) maps. So, in that frame work it's correct.
I'm sorry but here you lose me. In which local framework elements of [f,g] can be thought as part of a conjugacy class in the literal meaning of the term?
....
Quote: I do like equivariant, but that seems a tad restrictive when we're just considering one variable.
What do you mean?
....
I simply meant, when I think of this relation, I am usually assuming they are at least locally invertible; so the idea of conjugation is typically valid on some domain. But then again; that's because I always use holomorphic functions.
Secondly, equivariant seems to be a concept on flows; and this symbol seems independent of flows. The definition of equivariant seems reserved for flows. It kind of throws a stick in the mud when we just want a term for monoids.
Honestly, maybe "X"-Isomorphism would be good. They are isomorphisms... We'd just need a good term to put instead of "X."
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05/15/2021, 06:19 PM
(This post was last modified: 05/15/2021, 06:20 PM by MphLee.)
(05/14/2021, 09:27 PM)JmsNxn Wrote: Secondly, equivariant seems to be a concept on flows; and this symbol seems independent of flows. The definition of equivariant seems reserved for flows. It kind of throws a stick in the mud when we just want a term for monoids.
I still don't get what your trying to express. In my understanding: equivariance is more general than flows (if with flows you restrict yourself to continuous solutions to differential eqs.). This symbol is more general than flows in the strict sense but it is at the right level of generality when talking of monoids (it is a special case of the concept of natural trasformation).
Some instances of equivariance are \( f(\lambda x)=\lambda f(x) \), \( \chi(f^{\circ n}(x))=g^{\circ n}(\chi(x)) \) or \( e^{\lambda z}=(e^z)^\lambda \).
The real problem with equivariance is... how you call the sets [f,g]? Equivariant class? Class of equivariance? This doesn't sound right. Maybe set of equivariances of f and g.
MSE MphLee
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S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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It was established some years ago that other branches of tetration satisfy being the super of ln(x) + 2pi i or exp(x) + 2 pi i and similar results.
I believe Sheldon was first.
Is there any other thing about this branch hype of Leo that I missed ?
Not trying to be mean, but not sure why so much interest in it.
I mean it is either rather not interesting or not new as far as I can see.
It is just riemann surfaces ??
I do welcome and respect this new member though.
He clearly has insights and belongs here , no mistake.
Welcome Leo 
regards
tommy1729
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(05/15/2021, 11:25 PM)tommy1729 Wrote: It was established some years ago that other branches of tetration satisfy being the super of ln(x) + 2pi i or exp(x) + 2 pi i and similar results.
I believe Sheldon was first.
Is there any other thing about this branch hype of Leo that I missed ?
Not trying to be mean, but not sure why so much interest in it.
I mean it is either rather not interesting or not new as far as I can see.
It is just riemann surfaces ??
I do welcome and respect this new member though.
He clearly has insights and belongs here , no mistake.
Welcome Leo
regards
tommy1729
Hey, Tommy.
Different branches of tetration certainly do not do what you are saying. There are uncountably many solutions to the tetration equation; so I don't know what you mean by this.
This has to do with designing a classification of conjugate functions (or rather, we're trying to figure out what to call them first). Then we're trying to classify them. Or at least study their structure; in which we can consider them algebraically.
It's a very rich field, which is more in tune with abstract algebra; but you can do very quick manipulations if you have the set \( \[f,g\] \) be well defined. Such as modding out by this to form an equivalence relation; and then treating all elements in this set as one thing. But in order to do that, you need to understand the structure of it. And in what space it sits in which you can mod out. AT least, that's what I find so interesting about it. Mphlee is definitely more interested in the category aspect; where this relates very nicely to category theory.
Oh, Mphlee. I guess I misunderstood what equivariant means. I think this is the perfect term. I'll cast my vote for that. I thought it was reserved for,
\(
\chi(f(t,x)) = f(t,\chi(x))\\
\)
I didn't realize it could be null in the t variable.
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05/16/2021, 10:00 AM
(This post was last modified: 05/16/2021, 04:14 PM by MphLee.)
(05/15/2021, 11:25 PM)tommy1729 Wrote: [...] Is there any other thing about this branch hype of Leo that I missed ?
Not trying to be mean, but not sure why so much interest in it.
I mean it is either rather not interesting or not new as far as I can see.
It is just riemann surfaces ?? [...]
Hi Tommy, I apologize if the goal of this pool was not clear to you, my fault. As far as I'm concerned there is no hype on branches here. This post is NOT about branches. Is about naming conventions for superfunctions. If there was some misleading bit in the opening post I'd like to help clarify.
Here we are talking about defining a new unified terminological standard to refer to the solutions of Abel's, Scrhoeder's, Böttcher's and superfunction equations. I'm sure you agree with me that Abel functions, Schreoder functions, and supefunctions are very similar objects if we analyze them from an algebraic point of view. That's what I'd like to capture and it is indeed very interesting for some purposes.
So...no branches involved. The only link to Leo's thread is that he was referring to the set [f,g] I defined some months ago but generalized to the mutivalued/branched case.
Quote:[...]
I do welcome and respect this new member though.
He clearly has insights and belongs here , no mistake.
Welcome Leo
regards
tommy1729
I join you, of course he is welcome. Maybe was this post of yours intended to be a reply to his thread?
best regards
MphLee
ps: I hope you can leave an opinion on the pool Tommy! 
JmsNxn Wrote:And in what space it sits in which you can mod out. AT least, that's what I find so interesting about it. Mphlee is definitely more interested in the category aspect; where this relates very nicely to category theory.
Yes exactly! This is zero% formal or accurate: you can not mod out by those sets (because they are not disjoint) but from a philosophical point of view you would desperately want to mod out those sets obtaining an unique solution and you can certainly behave like your'e handling a modded out object.
on a note, maybe I'm repeating myself, but just by introducing those sets you are already doing category theory.
JmsNxn Wrote:Oh, Mphlee. I guess I misunderstood what equivariant means. I think this is the perfect term. I'll cast my vote for that. I thought it was reserved for,
\(
\chi(f(t,x)) = f(t,\chi(x))\\
\)
I didn't realize it could be null in the t variable.
OFC! If that equation is valid for all t then, as a corollary, also for t=1... which is our initial function. \( \chi(f(1,x)) = g(1,\chi(x))\\ \).
MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)
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(05/16/2021, 10:00 AM)MphLee Wrote: JmsNxn Wrote:Oh, Mphlee. I guess I misunderstood what equivariant means. I think this is the perfect term. I'll cast my vote for that. I thought it was reserved for,
\(
\chi(f(t,x)) = f(t,\chi(x))\\
\)
I didn't realize it could be null in the t variable.
OFC! If that equation is valid for all t then, as a corollary, also for t=1... which is our initial function. \( \chi(f(1,x)) = g(1,\chi(x))\\ \).
I guess that is pretty obvious  I guess I was assuming that \( \chi \) would depend on \( t \); but now that I think about it; the point is that it doesn't.
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yeah but it does not matter if we solve
f(2x) = exp(f(x))
or
g(x+1) = exp(g(x))
essentially those equations are equivalent.
And even though they might not hold everywhere , where they do also transfers when you change one into the other.
Instead of using new names, just write out the equations explicitly.
In particular if we want to publish a readable paper we should define things as clear as possible rather then making up new terminology or symbols.
Unless you find different methods for different solutions of those different morphisms I see no usefulness ?
I might be wrong about those branches ... Not sure what equations were valid there ... Maybe my memory plays tricks on me.
But If I may advertise one of my questions ; special cases of similar functional equations are still unsolved :
https://math.stackexchange.com/questions...ght2-1-fz2
That seems to be within the spirit of these functional equations.
Sorry for being hard on a new valuable member or posting this at the wrong place , but im surprised at such a poll.
regards
tommy1729
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