[note dump] Iterations and Actions

[MphLee]

While I was typesetting the crucial observation in the part 4 of my series of notes on iterations and actions I realized something very cool about it. The F-grade poetry in the title comes from the fact that this mysterious object puzzled me for many years and only yesterday I understood the real meaning of its morphisms.

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2022, May 13 - The tales of the legendary category \({\rm ACT}\) of Actions (idea)

Rememeber and action/iteration with time monoid \(A\) over a state space \(X\) is a binary function \(f:A\times A\to Y\) satisfying
\[f(0_A,y)=y,\quad f(a+_Ab)=f(a,f(b,y))\]
Think of this as a structure \((A,X,f)\).

Limitation There are two main ways to relate two iterations depending on their time monoid and on their state space.

1. The most important relations are the \(A\)-equivariances (state space relations/conjugations). These are possible only if the two iterations have the same time monoid;
   \[ (A,X,f)\overset{\varphi}{\to}(A,X,g) \quad\quad \forall a\in A,x\in X,\,\varphi(f(a,x))=g(a,\varphi(x))\]
   
2. The less known restriction/extension, I'll use the umbrella term divisibility relations, (time monoid relations/reparametrizations). These are possible only if the two iterations share the same state space (\(k\) satisfies \(k(a+_Ab)=k(a)+_Bk(b)\) and \(k(0_A)=0_B\));
   \[ (A,X,f)\overset{k}{\to}(B,X,g) \quad\quad \forall a\in A,x\in X,\,f(a,x)=g(k(a), x)\]
   

Equivariance relates the \(A\)-dynamics over two state spaces: it is possible to learn if they have the same periodic points, if one embeds in the others, or if we can represent part of a unknown dynamics by representing it as a known dynamics, like in the case of Abel and Schroeder equations (linearization). Equivariances tells us that while the sate \(x\in X\) evolves \(a\in A\) times the state \(\chi(x)\in Y\) evolves in the same way ("equivaries"). We can say equivariances are ways to deform the first dynamics in the second dynamics.
The biggest limitation is the apparent impossibility to have natural equivariance relations between two iterations using two different time monoids: we can only fix an time \(A\) and consider all of the iterations with that fixed time \(A{\rm -Act}\).

Question 1 A question that has always eluded me is: how the dynamics of a function \(f\) compares with the dynamics of its iterates \(f^n\)? What are the solutions of \[\phi (f^n(x))=f^{n+1}(\phi(x)) ?\]
Examples

• In general \(x\mapsto f^n(x)\) gets mapped to \(\chi(x)\mapsto g^n(\chi(x))\);
  
• When \(A=\mathbb N\) we just have superfunctions. \(0\mapsto n\) gets mapped to \(x_0\mapsto g^n(x_0) \);
  
• In the case of Schroeder functions we have a linearization. The evolution of an unknown map \(x\mapsto f^n(x)\) gets mapped to scalar multiplication \(\chi(x)\mapsto \lambda^n\cdot \chi(x) \).
  

Divisibility expresses the relation of two dynamics with different time over the same state space. This is about reparametrizing the orbits of a dynamics \(x_a=f(a,x)\in X\) get reparametrized by \(b\in A\) as \(x'_{b}=x_{k(a)}\in X\), i.e. \(g(b,x)=f(k(a),x)\). We can say we are globally extending or restricting the orbits of an iteration.
In this case the main limitation is: by knowing the divisibility relations alone there is not a clear way to known if it is possible for an iteration over a state space \(X\) to be the time extension, e.g. from integers to reals, of an iteration over a state space \(Y\). Everything we can do is: fix a set \(X\) and consider all of the iterations over \(X\) as bundles \({\rm Mon}/_{{\rm End}(X)}=\{f:A\to {\rm End}(X)\,|\, A\, is\, a \, monoid\}\).
Examples

• All the fractional iterations relations are relation of that kind. If \(f^2(x)=g(x)\) then for every \(n\) we have \(f^{2n}(x)=g^n(x)\) so we have a map \((\mathbb N,X, g)\overset{2\cdot}{\to}(\mathbb N, X,f)\);
  
• All the extension relations are of that kind. If \(i:\mathbb N\to \mathbb C\) is the inclusion of the naturals inthe complex then \(f(z,x)\) extends \(g^n(x)\) to the complex numbers iff for every \(n\) we have \(f({i(n)},x)=g^n(x)\) so we have a map \((\mathbb N,X, g)\overset{i}{\to}(\mathbb C, X,f)\);
  

QUESTION is it possible to have an unique place where we have all the various actions and study their interactions (time interactions and equivariant interactions) without caring if they have same time or same state space? In other words, is there an universal category \({\rm ACT}\) s.t:

1. contains all of the \(A{\rm -Act}\) for all the monoids \(A\) and
   
2. glues to the previous information also all the information of the functors \({\rm Ite}(-;Y) \)?
   

Is there a way to package all the information about equivariant maps (superfunctions ...) and the information of extension/restriction of the time?

Definition (ACT) Define the category of monoid actions \({\rm ACT}\):

• (Objects) are triples \((A,X,f)\) where \(A\) is a monoid, \(X\) is a set and \(f(a,x)\) is an \(A\)-iteration over \(X\);
  
• (Morphisms) \( (A,X,f) \overset{\boldsymbol \chi}{\to} (B,Y,g)\) are pairs \({\boldsymbol  \chi}=(k;\chi)\) where \(k:A\to B\) is a monoid morphism, \(\chi:X\to Y\) is a set function, s.t. \[\chi(f(a,x))=g(k(a),\chi(x))\]
  
• (Composition) of maps \( (A,X,f) \overset{\boldsymbol \chi}{\to} (B,Y,g)\overset{\boldsymbol \phi}{\to} (C,Z,h)\) is given by  \( (A,X,f) \overset{{\boldsymbol \phi}\circ {\boldsymbol \chi}}{\to} (C,Z,h)\) where \[{\boldsymbol \phi}\circ {\boldsymbol \chi}=(j;\phi)\circ (k;\chi)=(jk;\phi\chi)\]

Proposition The composition defined above is well defined.
Proof: assume that \(\chi(f(a,x))=g(k(a),\chi(x))\) and \(\phi(g(b,y))=h(j(b),\phi(y))\). We deduce from that \( \phi(\chi( f(a,y) ))=h( j(k(a)) , \phi(\chi(y))  ) \).
\[\begin{align}
\phi(\chi( f(a,y) ))&= \phi( g(k(a),\chi(y) ))\\
&=  g(j(k(a)),\phi(\chi(y)) ))\\
\end{align}
\]
Terminology Every morphism \({\boldsymbol  \chi}=(k;\chi)\) is made by the equivariant part \(\chi\) and the division part \(k\). We say the \(A\)-equivariances, e.g. superfunctions, are of purely equivariant nature and of form \(({\rm id}_A;\chi)\). The divisibility relations (re-parametrizations), e.g. extensions, are action maps  purely of "divisibility nature" \((k;{\rm id}_X)\).

Examples here some examples of action morpisms that are not of pure nature.

1. Consider the equation \(\chi(f(x))=f^{k}(\chi(x))\): this is a map \((m_k,\chi):{}(\mathbb N,X,f)\to(\mathbb N,X,f)\). Set \(X=\mathbb R\) and \(f=S\) the successor. Then \( (m_k;{\rm mul}_k):{}(\mathbb N,\mathbb R,S)\to(\mathbb N,\mathbb R,S) \) because \({\rm mul}_k(x+1)=S^k({\rm mul}_k(x))=k+{\rm mul}_k(x)\);
   
2. consider the action \((\mathbb N,\mathbb R\,S)\) and the action \((\mathbb R,\mathbb C\,f)\) the \(\mathbb C\)-action on the complex numbers \(f(z,w):=e^z\cdot w\). We have a map  \(  (\mathbb N,\mathbb R\,S)  \overset{\boldsymbol \chi}{\rightarrow}(\mathbb R,\mathbb C\,f) \) where \({\boldsymbol \chi}=(\ln b\cdot;\exp_b)\) because \[\forall n\in\mathbb N,\,x\in\mathbb R\, \exp_b(n+x)=e^{(\ln b) n}\exp_b(x)\]

Proposition every action map \(\boldsymbol \chi\) can be factored as a composition of its pure equivariant part and its pure divisibility part.