05/13/2021, 12:19 PM

This follows from the discussion held at: Jabotinsky IL and Nixon's program: a first categorical foundation, (May 10, 2021), Tetration Forum.

\( [f,g]:=\{\chi:X\to Y:\chi\circ f=g\circ\chi\} \)

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

(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 \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)