[note dump] Iterations and Actions

[MphLee]

2022, april 24 - actions, iterations and generalized elements PART 4 (reboot of old notes)

General recap \(A\)-iterations are well defined as monoid homomorphism from the time monoid to the monoid \({\rm End}(Y)\) of self-transformations of a state space \(Y\). Every \(A\) iteration defines what can be thought as a \(A\)-time evolution of the states \(Y\). Using the term iteration for those is suggestive and is meant to be an heuristic device forcing us to think of them as if they really are iteration of something and we wish that "something" to be a function \(f:Y\to Y\) even if we don't know if it is always possible.

• In the case of \(\mathbb N\)-iterations. we have a bijection to elements of \({\rm End}(Y)\);
  
• In the case of \(\mathbb Z\)-iterations we obtain only the bijective elements of \({\rm End}(Y)\);
  
• In the case of \({\mathbb Z}_2\)-iterations we get the involutions over \(Y\), i.e. particular elements of \({\rm End}(Y)\) that are inverse to themselves \(f^2={\rm id}_Y\);
  
• In the case of \(({\mathbb Z}_2,\cdot)\)-iterations we get the idempotents over \(Y\), i.e. particular elements of \({\rm End}(Y)\) s.t. \(f^2=f\);
  
• In the case of \(A=1\) then there exist only one \(1\)-iteration over \(Y\) and it is \({\rm id}_Y\in {\rm End}(Y)\);
  
• For general time monoids \(A\) we think of \(A\)-iterations as generalized elements in \({\rm End}(Y)\) or figures of shape \(A\) in \({\rm End}(Y)\);
  

It seems that, depending on the chosen monoid of time \(A\), the \(A\)-iterations corresponds to special self-transformations of the state space.
\[\begin{align}
{\rm Hom}(\mathbb N, {\rm End}(Y)) &\simeq {\rm End}(Y) \\
{\rm Hom}(\mathbb Z, {\rm End}(Y)) &\simeq {\rm Aut}(Y) \\
{\rm Hom}(\mathbb Z_2, {\rm End}(Y)) &\simeq {\rm Invl}(Y) \\
{\rm Hom}(1, {\rm End}(Y)) &\simeq 1 \\
{\rm Hom}(A, {\rm End}(Y)) &\simeq ???
\end{align}\]

Example in topology take a topological space \(X\), let \(\bullet={0}\) the trivial topological space on the point, \(I=[0,1]\) the real interval and \(S^1=\{(x,y):x^2+y^2=1\}\) the circle.

• Continuous functions \(P:{0}\to X\) are in bijection with points of \(X\). Call them figures of shape \(\bullet\) or just elements of \(X\);
  
• Continuous functions \(\gamma:I\to X\) are in bijection with paths of \(X\). Call them figures of shape \(I\) or just \(I\)-elements of \(X\);
  
• Continuous functions \(\gamma :{}S^1\to X\) are in bijection with loops in \(X\). Call them figures of shape \(S^1\) or just \(I\)-elements of \(X\);
  

To pursue (difficult) it is possible it makes sense of the previous scheme (points, paths, homotopies,..) in the context of iterations? Is it possible to find a a sequence of abelian groups \(1\to\mathbb N\to A_2\to A_3\to...\to A_n\to ...\) so that we can define something analogous of points, paths, homotopies, and so on but algebraically and define some kind of iterational homotopy theory of \({\rm End}(Y)\). Something like \({\bf \pi}_n(Y):={\rm ite}(A_n;Y)/{\simeq}\).

Definition (authentic time) every \(A\)-iteration defines a family of submonoid \(\{\varphi_A:\,a\in A\}\subseteq {\rm End}(Y)\). Call it \(A_\varphi\), a kind of quotient of \(A\) by \(\varphi\), we can see it as the authentic time of \(\varphi\).
It is the universal monoid making \(\varphi\) injective (faithful) in the iteration-time argument by factoring it.

If \(A\) is a commutative group \(A_\varphi=A/{\rm ker}\varphi\) is the quotient of \(A\) by the anihilator group of \(\varphi\).

Definition (extension) given an "extension" of time \(u:\mathbb N\to A\), or equivalently a choice of an element \(u(1)\in A\) (the unit of time), we obtain a "restriction" going from \(A\)-iterations to \(\mathbb N\)-iterations. Since the latter is equivalent to \({\rm End}(Y)\) we obtain the evaluation of the iteration at \(u(1)\in A\).

This what I'd like to call the underlying function map. This map answer to the question "what is \(\varphi\) iterating?"

\(\varphi\) is an \(A\)-iteration of \(u^*\varphi=\varphi_u\)

Observation 1 Introduce the language of preimages! The set of extensions of \(f\) is the preimage of \({\rm ev}_u: {\rm ite}(A;Y)\to {\rm End}(Y)\) at the value \(f\). It is the set of divisors \(\varphi\) , solution of the equation \[\varphi \circ u=f\]

Question (to do) Let's rework question 3 of this post. When the pre-image is empty? When is the restriction not surjective? Is it injective?

1. If \(u^*\) is injective and \(f\in {\rm im}u^*\) , then \(f\) can be extended uniquely to an \(A\)-iteration;
   The injectivity \(u^*\) amount to this property: \(u\in A\) is such that given two \(A\)-iterations \(\varphi,\psi\) then \[\forall y\in Y.\, \varphi_u(y)=\psi_u(y)=f(y)\quad{\rm implies}\quad \forall a\in A.\,\varphi_a(y)=\psi_a(y)\]
   
2. If \(u^*\) is sujective then \({\rm im}u^*={\rm End}(Y)\), i.e. every \(f:Y\to Y\) can be extended to an \(A\)-iteration;
   

The second point can be expressed with the "Wolfram-ian" slogan "All maps over \(Y\) have \(A\)-flows".

To pursue category-theoretically the solutions sets to those division problems (extension-restriction) are known to be functorial (see coslice/slice).

• The hom-set \(({\rm Mon}/_{{\rm End}(Y)})(f,\varphi)=\{u:\,f=\varphi\circ u\}\) is the set of \(u\in A\) such that \(f=\varphi_u\);
  
• The hom-set \((_{\mathbb N}\setminus{\rm Mon})(u,f)=\{\varphi:\,\varphi\circ u=f\}\) is the set of \(A\)-iterations \(\varphi\) extending \(f\), i.e. st \(\varphi_u=f\);
  

Denote \({\bf Ex}_u(f):=(_{\mathbb N}\setminus{\rm Mon})(u,f)=\{\varphi :\,\varphi_u(y)=f(y)\}\), the set of extensions of the integer iteration of \(f\) to \(A\) time-iterations that exhibit \(f\) as \(\varphi_u\). How this set relates to the solutions sets of the functional equation \( \chi(u+a)=f(\chi (a)) \).

Proposition 1 For every \(y_0\in Y\) and every \(\varphi \in {\bf Ex}_u(f) \) exists a \({\hat \varphi}_u\in\{\chi:A\to Y\,| \, \chi\lambda_u=f\chi \} \)
\[{\hat -}:{\bf Ex}_u(f)\times Y\to{\rm Hom}_{\mathbb N{\rm -Act}}(A^\lambda, Y^f)\]
Proof: we demonstrate that every solution to the iteration problem defines, for every choice of a base point \(y_0\), a solution to a superfunction problem.
Assume we have an action \(\varphi_a(y)\)s.t. \[\varphi_{a+b}(y)=\varphi_{a}(\varphi_{b}(y));\quad \varphi_0(y)=y\quad{\rm and}\quad \varphi_u(y)=f(y)\]
Then, for every \(y_0\in Y\) we define a map \({\hat \varphi}_{y_0}:A\to Y\) \[{\hat \varphi}_{y_0}(a):=\varphi_a(y_0)\] we prove we obtain an \(A\)-equivariant map: \[\forall a,b\in A,\,{\hat \varphi}_{y_0}(a+b)=\varphi_a({\hat \varphi}_{y_0}(b))\]
By setting \(a=u\): \(\,{\hat \varphi}_{y_0}(u+b)=f({\hat \varphi}_{y_0}(b)) \).  \(\quad \square\)

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FRACTIONAL ITERATION, DIVISIBILITY AND FUNCTIONAL LOGARITHM

Observation 2 assume we have two \(\mathbb N\)-iterations, i.e. two maps \(f(y)\) and \(g(y)\), what it means for \(f\) to be a fractional iterate of \(g\)? In symbols: \[f(f(y))=g(y).\]

• Now, remember the bijection \({\rm End}(Y)\simeq {\rm ite}(\mathbb N;Y)\). If we see the phenomenon at the level of \({\rm End}(Y)\) it says: \(f\) is a frac. iterate of \(g\) can be translated in the proposition \(\exists k: f^k=g\);
  
• If instead we see it at the level of \({\rm ite}(\mathbb N;Y)\) we obtain a factorization problem: \(f\) is a frac. iterate of \(g\) is expressed by the proposition "\(\exists k\) such that for every \(n\) \({\rm ite}_f(k\cdot n)={\rm ite}_g(n)\)"..
  

The map \(m_k:\mathbb N\to \mathbb N\) sending \(n\mapsto k\cdot n\) is an monoid automorphism (distributivity): this produces a factorization

Observation 3 (The Power Action)This factorization is deeply linked with the lattice of integers under divisibility and contains a lot of information about the multiplicative structure of the integers (see the crucial observation in this post), hence of extensions to rational numbers. All of this information can be packed naturally into a single algebraic gadget that I call "the power action of \(A\)": the action of \( {\rm End}_{\rm Mon}(\mathbb N, +)\) on \({\rm ite}(A;Y)\).

Proposition 2 every endomorphism \(k:A\to A\) of the time monoid \(A\) induces a a map "raising" every \(A\)-iteration to the \(k\)-th power and the result is still an \(A\)-iteration.

Proof: functoriality of \({\rm ite}(-;Y)\) induces a morphism of monoids, thus an action \[{\rm pow}(j\circ k,\varphi)={\rm pow}(j,{\rm pow}( k,\varphi))\]
call this the \(k\)-power map, where \(k:A\to A\) is a monoid morphism. \(\square\)

Corollary 2 if \(A=\mathbb N\) then  \( {\rm End}_{\rm Mon}(\mathbb N, +)\simeq (\mathbb N,\cdot)\) is the mutiplicative monoid of natural numbers. In fact \[\forall k,j\in\mathbb N.\, f^{jk}=(f^k)^j\]

CLAIM We can use this action to: a) formally define a kind of "functional logarithm" computing naturally for some functions \(\alpha:Y\to Y\) commuting with \(f:Y\to Y\), i.e. \(f\alpha =\alpha f\), their rational height \({\mathfrak {log}}_f(\alpha)\in\mathbb Q\); b) prove this map always exists and; c) find its domain of definition (what I call the intrinsic iterates of \(f\)).
\[{\mathfrak {log}}_f(\alpha)=\frac{n}{d}\in\mathbb Q\quad iff \quad \alpha^d=f^n\]