Hi, I'm sorry I don't have the energy to follow all of this properly.

But I have a lil bit of time so I offer some thoughts.

About missing things... I believe it's a problem everyone has, and members of the forum in particular since nobody here is paid to do research and we are forced to grapple with these problems only with a skinny part of our time/life force. Sad but true.

Another coin I want to offer is a terminological one. I notice that often in this forum some topics are faced by only few viewpoints and that the purely algebraic side is often ill developed.

It is increasingly more evident for me that once the right framework is established all the continuous; discrete; matrix e and integral; topological and analytical phenomena we are separately dealing with will merge into a well connected big picture. Obviously, to do this, much more culture, time and coordinated research effort is needed, or just the intervention of a erudite professional with a interest in this field.

Back to the point... much use of the term "semigroup", "homomorphism" and "actions" has been made and often in a superficial, inprecise, manner. I'd like to give my coin making this more precise, again.

A semigroup is a quite abstract gadget: a semigroup \((T,*_T)\) is a set of things that is closed under an associative, non necessarily commutative, operation: stop nothing more nothing less... no functions, no way to evaluate elements. Define what is an action and what does it mean to act for a semigroup. A semigroup \(T\) can act over other mathematical objects such as spaces, say \(X\) is a space-like object, by associating to each element \(t\in T\) a self-transformation of \(\phi_t: X\to X\) that is compatible with the spacial structure of \(X\) such that composing two consecutive transformations associated witht wo elements of \(T\) $$X\overset{\phi_s}{\longrightarrow}X \overset{\phi_t}{\longrightarrow} X$$ is the same as acting on \(X\) by the spatial transformation associated with \(t*s\) $$X\overset{\phi_{t*s}}{\longrightarrow}X.$$ This is a quite general scenario.

It is important to have clear in mind that an action of a semigroup \(T\) over a space-like object \(X\) is equivalent as giving a semigroup homomorphism from that semigroup \((T,*_T)\) to the semigroup \(({\rm End}(X),\circ)\) of space-preserving self-transformation of that space-like object. Since in our intuition \(T\) is the domain of of time-like object/instants, that amounts to specifying a time evolution of the things that lies in \(X\), that is a \(T\)-dynamics.

$$\lbrace \, T -{\rm actions\, over} X\, \rbrace \cong \lbrace \, {\rm semigroup\, homomorphisms\, from} \, T {\rm to \, End}(X) \rbrace$$

i.e. in technical terms that the following set are in bijective correspendence

$$\lbrace \, T\times X\overset{\varphi}{\longrightarrow} X\, |\, {\rm s.t.}\, \varphi(t*s,x)=\varphi(t,\varphi(s,x)) \rbrace \cong \lbrace T \overset{\phi}{\longrightarrow} {\rm End}(X) \, |\, {\rm s.t.}\, \phi(t*s)=\phi(t)\circ \phi(s) \rbrace$$

When we say that a function \(f:X\to X\) or a matrix \(M:V\to V\) acts over something we actually mean, precisely, that there is a semigroup \(T=(\mathbb N,+)\) that is acting on that space by a morphism of semigroups that satiesfies \(\phi_1=f\) of \(\phi_1=M\): in that case we are justified to call it integer iteration.

Concrete Examples
A (scaling in abelian groups) - Notable example of this are vector spaces: let \(k\) be a field and \(V\) a vector space with scalars living in the field \(k\). Let\(T=(k^\times,\cdot ) \) be the monoid structure of the field under multiplication. Take our spatial object to be the abelian group \((V,+)\) of vector. Then \(k^\times\) acts over \(V\) by abelian group morphisms \(\mu_t:V\to V\) for \(t\in k\) a scalar in the field, i.e. transformations of vectors that respect the vector addition. It is an action by scalar multiplication and respecting the vector addition means we have distributivity: $$(t\cdot s){\bf v}=t(s{\bf v})$$ we can interpret it as scaling by a factor but also by linear motion of a point running away from the origin...i.e. drawing the linear subspace \(\langle \bf v\rangle\) so in a sense it's dynamical.

B1 (matrices) - A second related example is the case of a group of matrices acting over vector spaces. Let \({\sf GL}_n(\mathbb R)\) be the group of \(n\)-th order invertible square matrices with real coefficients and let \(V\) be an \(n\) dimensional vector space. By linear algebra we know that, fixing a base \(\mathcal B\) of \(V\) gives us a vector space isomorphism, i.e. bijective linear application $$\varphi_{\mathcal B}:\mathbb R^n \overset{\cong}{\longrightarrow} V$$ That isomorphisms, in turn, let us describe every linear operator over \(V\) as an \(n\)-square matrix, i.e. the isomorphism lifts to an isomorphism of the respective monoid of operators $$ {\sf M}[\varphi]: {\sf M}_n(\mathbb R) \overset{\cong}{\longrightarrow}{\rm End}_{\bf Vec}(V)$$

This map sends each matrix \(A\) to the operator \( {\sf M}[\varphi]_A({\bf v}) = \varphi(A\cdot \varphi^{-1}({\bf v})) \). It also sends matrix multiplication to composition of linear operators.

Notice now that the group \({\sf GL}_n(\mathbb R)\) is a submonoid of the monoid \({\sf M}_n(\mathbb R)\) of all \(n\)-th order square matrices. This inclusion is a map that respect the matrix multiplication, hence a monoid morphism: call it \(i\) $${\sf GL}_n(\mathbb R) \overset{i}{\longrightarrow} {\sf M}_n(\mathbb R).$$

At this point it is enough to compose the monoid morphisms to obtain a morphism of the general linear group over the vector space \(V\).

$${\sf GL}_n(\mathbb R) \overset{i}{\longrightarrow} {\sf M}_n(\mathbb R) \overset{\cong}{\longrightarrow}{\rm End}_{\bf Vec}(V)$$

In this way we have a group \({\sf GL}_n(\mathbb R) \)-action of invertible matrices acting over a vector space \(V\) by linear operators.

B2 (rotations) Consider the group isomorphism sending sum of angles to multiplication of

unitary complex numbers \(e: (\mathbb R,+)\to \mathbb T\): \(\theta\mapsto e^{i\theta}\). Is is known that there is a group isomorphism morphism sending complex numbers to \(2\times 2\)-matrices. In particular this map sends bijectively all the unitary complex numbers \(z\in\mathbb T\), the ones with \(|z|=1\) to rotation matrices in \({\sf SO(2)}\). $$\mathbb T \overset{R}{\longrightarrow} {\sf SO}(2)\,\,\,\,\, R(e^{\theta i}):=

\begin{bmatrix}

\cos \theta & -\sin \theta \\

\sin \theta & \cos\theta \\

\end{bmatrix}$$ Post-composing both with the injective morphism \( {\sf SO}(2)\overset{j}{\longrightarrow} {\sf M}_2(\mathbb R)\) exhibiting the special orthogonal group as a submonoid of the \(2\times 2\)-real valued matrices we obtain an action of the additive group of real numbers over the euclidean plane, by rotation, and action that decomposes in several intermediate actions.

$$\mathbb R \overset{e^{i\cdot}}{\longrightarrow} \mathbb T \overset{R}{\longrightarrow} {\sf SO}(2)\overset{j}{\longrightarrow} {\sf M}_2(\mathbb R)\cong {\rm End}_{\bf Vec}(\mathbb R^2)$$

We can see this as time parametrizing continuous rotations of a vector \(R_{\theta+\tau}({\bf x})=R_{\theta}(R_{\tau}({\bf x}))\).

C (representation theory) - It is evident by now that given any group \(G\) and a group homomorphism \(\phi: G\to {\sf GL}_n(\mathbb R) \) we obtain, by default a linear action of \(G\) over the space \(\mathbb R^n\). This is exactly the topic of representation theory: considering all the ways to "linearize" a group, i.e. in a sense, all the ways of

representing \(G\) multiplication as the multiplication over vectors \((gh){\bf v}=g(h{\bf v})\) and \(g({\bf v+u})=g{\bf v}+g{\bf u}\) or, equivalently, all the ways in which, given an sum-like operation, we can extend its iteration to \(G\).

A

Functorial Quantum field theory (FQFT), under the

Schrödinger picture of QM dual to the Heisenberg picture, whatever this means, instead can be thought about

representing wordlines (cobordisms of manifolds), instead of just time continua, as linearly acting over hilbert spaces. In other words, a (functorial) quantum field theory is a linear representation of a space-time manifold acting of infinite dimensional hilbert spaces by evolution operators, or something along those lines.

D (Matrices methods) - it seems to be that the trick here is done by taking the space of formal series as a monoid \(\mathbb R[[X]]\) and representing it as infinite matrices by the Bell and Carlemann machines, i.e. a monoid morphism \({\sf C}:\mathbb R[[X]]\rightarrow {\sf M}_{\infty}(\mathbb R)\). Then if for \(f\in \mathbb R[[X]]\) the associated matrix \({\sf C}[f]\) would give powers related to the iteration in the monoid \(\mathbb R[[X]]\). This process should be analyzed ALSO from this point of view imho...

E (iteration in general) - but then, after all these examples, is it enough to have an action of a semigroup, eg. \(T=(\mathbb R,+)\), \(T=(\mathbb C,+)\) over a space of things \(X\), i.e. a map sending \(t\in T\) to \(f_t:X\to X\), to be able think of \(f_{\cdot }:T\to {\rm End }(X) \) as having to do with the iteration of something? I think that the action, to have the right to be called iteration be should at least be a monoid action. So \(T\) must be a monoid and not only a semigroup. There must exist something \(0_T\in T\) that gives us the identity \(f_{0_T}={\rm id}_X\), i.e. the do-nothing action.

Is it enough? Ofc... this could be called algebraic iteration... but in our intuition \(T\) must have some time-like features... like being a topological space... the action must be continuous... so continuous iteration is a continuous action of a topological monoid \(T\)? Do time has to be invertible? Commutative? Should we call an action an iteration only if \(T\) is a topological abelian group? Then iteration theory would be limited to actions of abelian Lie groups.

Notice that those actions define vector fields so we reconnect with classical dynamics.

If instead of semigroup actions we restrict to monoid actions, it turns out that spectral analysis and the generalized version of spectra, periodic points, eigenstates, eigenvalues and so on can be defined in the case of arbitrary monoid actions at the level of functor categories, as Lawvere explains in

Taking Categories Seriously, 2005, TAC.

F (near-semirings geometry?) What if is the right way to think about iteration is not as an action over a space-like object but as an action over the trasformations of a space-like object... i.e. as an action over a monoid? The set of endofunctions \(M^M\) of a monoid \(M\) is itself a monoid under composition but it also inherits a monoid structure by pointwise \(M\)-multiplication: both structure interacts only by left distributivity \((\psi\chi) \circ \varphi=\psi\varphi \circ \chi\varphi\) giving us a structure of right near-semiring. Finding a way to truly exponentiate elements of \(M\), e.g. let it be functions, amounts to defining a near-semiring morphism \({\mathfrak f}:\mathbb R\to M^M\). Let \(s,t\in \mathbb R\) and \(m\in M\), being an unitary near-semiring morphism translates in the following equations holding $${\mathfrak f}_0(m)=0_M;\,\,\,{\mathfrak f}_{t+s}(m)=({\mathfrak f}_{t}{\mathfrak f}_{s})(m)={\mathfrak f}_{t}(m){\mathfrak f}_{s}(m)$$

$${\mathfrak f}_1(m)=m;\,\,\,{\mathfrak f}_{ts}(m)=({\mathfrak f}_{t}({\mathfrak f}_{s}(m))$$

Maybe before asking for how to iterate naturally some process we should invest more in the question: what does it means to iterate and what it means to naturally iterate?