[note dump] Iterations and Actions

[MphLee]

Damn, Mphlee, I hope this makes sense....

Moved this discussion HERE (link).

Here I begin to save some old pages, illustrations and notes that later evolved into the present thread.
This is part 1 of the genesis of the idea of intrinsic iterates.

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2015, September - The lattice of divisibility and fractional iterates

The integers are ordered into a lattice, the lattice of the divisor relation. The lattice operations are \(\rm gcd\) and the \(\rm lcm\). Here a fragment of how it looks

^IMG

lcm works in this lattice as vector addition would work, instead integer multiplication is always greater than that.

Question how to detect when a function behaves as a fractional iterate? What iscount as a fractional iterate? Let's simplify the problem.
A root of \(f:X\to X\) is a function \(g:X\to X\) that solves \[g^m=f\].

Consider for each function \(f:X\to X\) the set \(\langle f\rangle:=\{f^0,f^1,...,f^n,...\}\). The map associates to every function the submonoid generated by that element.
Proposition. \(g\) is a root of \(f\) is a relation that is homomorphic to the dual inclusion of submonoids: \(g\) is a root of \(f\) iff \(\langle f\rangle \subseteq \langle g\rangle\).
Proof. let \(g^m=f\). Take \(\alpha \in \langle f\rangle\) then \(\alpha=f^k=(g^m)^k= g^{km}\) thus \(\alpha \in \langle g\rangle\). If \(\langle f\rangle \subseteq \langle g\rangle\) then \(f\in \langle g\rangle\). By definition we conclude that exists an \(m\) st \(f=g^m\), i.e.  \(g\) is a root of \(f\). \( \square\)

Let \({\bf P}\subseteq {\rm Sub}({\rm End}(X))\) be lattice of submonoids of the monoid \({\rm End}(X)\) generated a single element , i.e. of the form \(\langle f \rangle\). The lattice order is given by inclusion, lattice and meet are given by intersection of submonoids and the direct product. Fix a function \(f\). Consider the sublattice of all the submonoids generated by all its integer iterates: call it \({\bf L}^+_f\subseteq {\bf P}\). It looks like this: 

Observe how the direction of the lattice order is reserved respect to the divisibility relations of the integers.
\[\begin{align}
m &| n&& {\textrm m\, divides \, n} & \exists k,\,m\cdot k&=n\\
g &\preccurlyeq f&& {\textrm g\, root\, of\, f}  & \exists k,\,g^k&=f\\
\langle g \rangle &\supseteq \langle f \rangle&&  & \exists k,\,g^k&=f
\end{align}\]

The information of the rational iterations (the roots) is encoded in the lattice of submonoids. We can in fact extend the lattice \({\bf L}_f^+\) by adjoining all the submonoids that contains some iterate of \(f\): this process is showed in the next picture as the process of prolonging the lattice under \(f\).

Definition. Define the lattice of roots of \(f\) as \({\bf L}_f^-\subseteq {\bf P}\) as \[{\bf L}_f^-:=\{\langle g\rangle \in {\bf P}\,|\, \langle f \rangle\subseteq \langle g \rangle\}\]

Philosophy. ideally we can think of an infinitesimal generator of \(f\) as the smallest iterate \(\lim_{\delta\to 0}f^\delta\) of \(f\) that generates all the rational/real iterates, i.e. \(\langle f^{m/n} \rangle\subseteq \langle f^\delta \rangle\).