[Question] Classifying dynamical system by connected components

[MphLee]

If we consider all the structures of the form \(X=(X,f)\) where \(f:X\to X\) is an endofunction of the set \(X\), so no topology, continuity, and so on, we have some interesting classification going on. I'm not going to prove anything, but it all follows from the fact that such systems \(X\) are objects of the category of discrete time actions \(\mathbb{N} {\rm Set}\), called autonomous time discrete dynamical systems.

Using this framework, we can approach discrete dynamics in a more visual and geometric manner. We can classify systems in a way very reminiscent of how we can classify vector spaces or other geometric objects.

Connected dynamical systems 
We can ask if \(X\) is connected. This happens if given two states of the system \(x,y\) there exists a third state \(z\) that connects them in the dynamics, i.e., such that \(f^n(x)=z=f^m(y)\). Connected dynamical systems can be visualized as made by only a single piece. Some examples of connected systems are:

• \( \bar { \mathbb N}=(\mathbb N, S) \) the naturals with successor;
  
• \( \bar { \mathbb Z}=(\mathbb Z, S) \) the integers with successor;
  
• \( \bar { \mathbb Z}_m=(\mathbb Z_m, S) \) this is the \(m\) element set with successor modulo \(m\), i.e., \(S(n):=n+1 \,{\rm mod} \,m\);
  
• \( \bar { \Omega}=(\mathbb N, P) \) this is the natural number with cut-off predecessor, i.e., \(P(n)={\rm max}(0,n-1)\);
  
• for \((p,h)\in \mathbb N^+\times \mathbb N\) define \( \bar { \mathbb N}_{(p,h)}=(\mathbb N, S) \) as \(p+h\) element set \(\{0,1,...,h, h+1,...,h+p-1\}\) with dynamic \(S(n)=n+1\) and \(S(h+p-1)=h\);
  
• \(K=(\mathbb N, 0\cdot)\) is the constant function with \(0\) as the only fixed point where every \(n\) is mapped to \(0\);
  
• \((\mathbb N[x],\frac{\rm d}{{\rm d}x})\) where the formal polynomial with natural coefficients \((a_0,a_1,a_2,...,a_d)\) is mapped to its formal derivative \((a_1,2a_2,3a_3,...,na_n)\). This dynamical system has a single fixed point.
  

Non-connected dynamical systems 
A system that is non-connected is the union of a finite or an infinite number of connected dynamical systems. Denote with \(\pi_0 X\) the set of connected components of \(X\); it is a partition of \(X\) into equivalence classes by the relation of "connected states." If \(\|\pi_0X\|=1\), then we have a connected dynamical system. We can see the ordinal \(\beta_0(X)=\|\pi_0X\|\) as a kind of Betti number of the dynamical system. This means that for each dynamical system \(X=(X,f)\), we can find a unique decomposition 
\[\boxed{X=\bigsqcup_{\lambda\in \pi_0X} X_\lambda}\]
as a sum of connected dynamical systems \(X_\lambda\). We have the following decompositions of non-connected dynamical systems:

• Let \(\bar{\mathbb R}=(\mathbb R,S)\) be the real numbers and the successor function.
  
• Let \( h_{1,b}=(\mathbb N, h_{1,b}) \) be the naturals with addition by \(b\), i.e., \(h_{1,b}(n)=b+n\);
  
• Let \( h_{2,b}=(\mathbb N, h_{2,b}) \) be the naturals with multiplication by \(b\), i.e., \(h_{2,b}(n)=b\cdot n\);
  
• Let \( {\rm Collatz}=(\mathbb N, f) \) be the naturals with the Collatz map.
  

Proposition: 
i) First of all, the Collatz dynamic is connected, \(\|\pi_0 ({\rm Collatz})\|=1\) iff the Collatz's conjecture is true.
ii) For the real numbers \(\pi_0\bar{\mathbb R}\cong [0,1)\) and since each real number can be decomposed in integer and fractional part \(r,n\) because \(x=r+n\) for \(r\in[0,1)\) we decompose the dynamics as \[\bar{\mathbb R}\cong [0,1)\bar{\mathbb N}\]
iii) Then \(\|\pi_0 (h_{1,b})\|=b\) if \(b\neq 0\) \[h_{1,b}\cong b \bar { \mathbb N}\] each state is decomposed uniquely as \((r,h)\) where \(n=r+bh\) with \(0\leq r < b\) (this is just the Euclidean division). 
iv) And \(\|\pi_0 (h_{1,b})\|=\aleph_0\) if \(b=0\) since we have the dynamic of the identity where each state is a fixed point and \(h_{1,0}\cong \aleph_0 \bar { \mathbb Z}_1\). 
v) For rank 2, i.e., multiplication, we have \(h_{2,0}=K\) thus \( \|\pi_0 (h_{2,0})\|=1\); \(h_{2,1}=h_{1,0}\) is the identity and \(\|\pi_0 (h_{2,b})\|=\aleph_0\) has infinite connected components if \(b\geq 2\). In the infinite case, we have the decomposition \[h_{2,b}\cong 1 \sqcup \mathbb N[x] \bar { \mathbb N}\] this decomposes each number \(n\) as the pair \((r,h)\) where \(b\) does not divide \(r\) and such that \(n=r\cdot b^h\).

Motivation
Why this matters? Because if let \({\rm hom}(X,Y)\) be the set of dynamical system maps \( \phi : (X,f)\to(Y,g) \), or equivarian maps with \( \phi(f(x))=g(\phi(x)) \) then

• \({\rm hom}(X,\bar { \mathbb N})\) is the set of positive integer valued abel functions of \(X\);
  
• \({\rm hom}(X,\bar { \mathbb R})\) is the set of real valued abel functions of \(X\);
  
• \({\rm hom}(X,h_{2,b})\) is the set of positive integer valued schroeder functions of \(X\);
  
• \({\rm hom}(\bar { \mathbb N},X) \) contains the orbits of \(X\);
  
• \({\rm hom}(\bar { \mathbb Z}_1,X) \) contains the all fixed points of \(X\), also called equilibria;
  
• \({\rm hom}(\bar { \mathbb Z}_n,X) \) contains the all n-periodic points of \(X\);
  
• \({\rm hom}(\bar { \mathbb R},X) \) contains the all the possible real superfunctions of \(X\), continuous, analytic and disconitnuous;
  
• \({\rm hom}(\bar { \mathbb R},h_{3,b}) \) contains the all the possible real superfunctions of exponentiation, also the tetration;
  

If we know the decomposition in connected components of \(X=\bigsqcup_{\lambda\in \pi_0X} X_{\lambda}\) we can derive a parametrization of the hom-sets since
\[\boxed{ {\rm hom}(\bigsqcup_{\lambda\in \pi_0X} X_{\lambda},Y)\cong \prod_{\lambda\in \pi_0X}{\rm hom}(X_{\lambda},Y) }\]

Question 
What is known about \(\pi_0 f\) for \(f\) being the logarithm or the exponentiation over the real numbers? Over the complex numbers?

[MphLee]

Just a Note, the last formula is important.
I have some refinement of it, some old formulas and some new ones.
The basic philosophy of this is that bigger superfunctions can be decomposed into union of smaller superfunctions.

Soon Ill add the refined formulas.

[MphLee]

[warning: my last claim about the surjective map was wrong, I was in a hurry and made a dumb mistake about epi inducing mono, and colimits turning into limits. I hope now the claim is not obviously wrong]

In the following I'll use losely the term superfunction, as a synonim for equivariant map. This is just because the term superfunction is familiar to the forum users, what I really mean is an equivariant function, i.e. a function that sends iterations to iterations.

Here is the full formula. Here we assume \(X, Y\) are \(\mathbb N {\rm Set}\), i.e. discrete dynamical systems and \(X=\bigsqcup_{\lambda\in \pi_0X} X_{\lambda}\) and \(Y=\bigsqcup_{\theta\in \pi_0Y} Y_{\theta}\) their decompositions into, smaller, connected dynamical sub-systems. This Euclidean-algorithm-like decomposition is the essence of all the following consequences. Given that, the superfunctions from \(X\) to \(Y\) decompose into vectors of smaller superfunctions.

\[ {\rm hom}(X,Y)\cong \prod_{\lambda\in \pi_0X}\bigsqcup_{\theta\in \pi_0Y}{\rm hom}(X_{\lambda},Y_{\theta}) \]

This formula is obtained by the decomposiion of \(X\) and \(Y\) using the following two theorems: 

• A superfunction \(f:X\sqcup Y\to A\) is equivalent to giving two superfunctions \(f_0:X\to A\) and \(f_1:Y\to A\).  This is in symbols represented as \({\rm hom}(X\sqcup Y, A)={\rm hom}(X, A)\times {\rm hom}(Y,A)\);
  
• if \(X,A,B\) are connected dynamical systems then  a superfunction \(f:X\to A\sqcup B\) take values into \(A\) OR into \(B\), never in both. In symbols \({\rm hom}(X,A \sqcup B)={\rm hom}(X,A )\sqcup{\rm hom}(X,B)\) .
  

REFINEMENT BY ORDER/PERIODIC POINTS
We can refine this even more in different ways. In 2022 I believed that the best way to refine the formula was to notice that a bunch of sets \( {\rm hom}(X_{\lambda},Y_{\theta}) \) are just empty. They are empty every time the order of \(\theta\) is not a divisors of the order of \(\lambda\). Here the order of a connected dynamical system is defined to be the order of its elements/states. A state \(x\in X\) has order \({\rm ord}_X(x)=k\) if and only if \(f^n(x)\) is a \(k\)-periodic point for some sufficiently large \(n\), the order is \(0\) if \(f\) has not periodic points. If the system \(X\) is connected then every state \(x\in X\) has the same order \(=k\). Call that the order of \(X\), i.e. \({\rm ord}(X)=k\).
Now recall that \(X\) always admit a decomposition in connected components. Since each state belonging to the same component has the same order, the order map \({\rm ord}_X:X\to \mathbb N\) factors via the map \(\epsilon_X:X\to \pi_0 X\), yielding the map that gives to each connected component \(\lambda \in \pi_0 X\) a definite natural number, its order \(o_X(\lambda)\in \mathbb N\).

As a rule of thumb, we have that \( {\rm hom}(X_{\lambda},Y_{\theta}) \) is non-empty only when \(o_Y(\theta)| o_X(\lambda)\). There are, however, two complexities since we have to define the divisibility properties of \(0\) and because dynamical systems of order \(0\), what we have called as "aperiodic", comes in two kinds: eternal dynamics, where there are infinite past states of the systems, and non-eternal ones, or generated, i.e. there are zero-time/beginning states and all the future states come from those. We divide aperiodic dynamics of order into two types, eternal \(\infty\) and generated \(0\).  These are the rules, that can be easily proved:

\[\begin{array}{|c||c|cc|}
\hline
& periodic &aperiodic \\
{\rm dom \to \rm cod} & k&\infty& 0\\
\hline
\hline
j&\neq \varnothing\,\,{\rm iff}\,\,k|j &\varnothing &\varnothing \\
\hline
\infty &\forall k.\,{\rm non-empty}&{\rm non-empty}&\varnothing \\
\hline
0 &{\rm non-empty}&{\rm non-empty}&{\rm non-empty} \\
\hline
\end{array}\]

This means that the divisibility relation must be extended as follows

• if \(k\neq 0\), \(0\) and \(\infty\) do not divide \(k\); Thus periodic dynamics can not be sent into a-periodic ones.
  
• if \(k\) is finite then \(k\) divides \(\infty\) but \(0\) doesn't divide \(\infty\); Aperiodic eternal dynamical systems can be rolled up into periodic dynamics but not into aperiodic non-etrnal, because there is not room there for all the past states;
  
• both finite (\k\)s and \(\infty\) divide \(0\), i.e everything divides \(0\); This means that aperiodic generated dynamics can map always into other dynamics.
  

The above defined extended divisibility relation on \(\mathbb N\cup \{\infty\}\), where \(1 \,|\, k\,|\,\infty \, | \,0\), defines a binary relation on \(\pi_0 Y\times \pi_0 X\). We say that a component \(\theta \in \pi_0 Y\) divides \(\lambda \in \pi_0 X\), \(\theta |\lambda\) iff it's order divides the other's one \(o_Y( \theta)\, |\,o_X ( \lambda)\). At this point we can just define the set of relative divisors of a component \(\lambda \in \pi_0 X\) as the set of components \(\theta \in \pi_0 Y\) whose order divides \(\lambda\)'s.

\[{\bf D}_{X,Y}(\lambda):=\{\theta \in \pi_0 Y\, :\, \theta | \lambda \}\]

Using these sets we get a refinement of the previous scomposition

\[\boxed{ {\rm hom}(X,Y)\cong \prod_{\lambda\in \pi_0X}\bigsqcup_{\theta\in {\bf D}_{X,Y}(\lambda)}{\rm hom}(X_{\lambda},Y_{\theta}) }\]

And this means that knowing the the periodic points of your dynamics is not enough to fully characterize it, and its relationship to other dynamics, because one needs to know aperiodic eternal and aperiodic generated components too.

REFINEMENT VIA HEIGHTS/ DENSITY THEOREM
By the density theorem, all presheaves are colimit of representables, we know that that each connected dynamical system is a colimit of representables dynamical systems. Since there is only one representable dynamical system, i.e. \(\mathbb N\) with the successor of natural numbers, we have that each connected dynamics \(X\) is actually made up of a bunch of copies of \(\mathbb N\) glued together.

\[X\cong {\rm colim}_I \mathbb N\]

This means that locally each dynamics looks like the successor of natural numbers, and from here all the difficulties in extending discrete time dynamics to continuous time dynamics.
I claim that there is a preferred, almost canonical, presentation of a connected dynamical system as a colimit of many copies of \(\bar{\mathbb N}_{(p,h)}\) and of \(\bar{\mathbb N}_{(p,\infty)}\) glued together somewhere, and with the period \(p\) fixed. Recall, from the first post, that \(\bar{\mathbb N}_{(p,h)}:=\bar{\mathbb N}/_{h+p\mathbb N}\) is q quotient of the naturals, thus a colimit.

Claim: let \(|{\rm gen}(X)|=g_X\) be the cardinality of generating states, i.e. generators, of \(X\) and \(|{\rm ete}(X)|=e_X\) the cardinality of "filaments" of states going back to the infinite past, into the "negative eternity". Then \(X\) is the colimit of \(g_X\) copies of \(\bar{\mathbb N}_{({\rm ord}(X),h)}\), for various heights \(h\), and of \(e_X\) copies of \(\bar{\mathbb N}_{({\rm ord}(X),\infty)}\), glued together at their intersection \(V_X={\rm generated}\cap {\rm eternal}\).

\[X\cong {\rm colim}_{g\in {\rm gen}(X)} \bar{\mathbb N}_{({\rm ord}(X),{\rm ht}_X(g))} \cup_{V_X} {\rm colim}_{e\in {\rm ete}(X)} \bar{\mathbb N}_{({\rm ord}(X),\infty)}\]

This, given \(Y\) connected induces, by the universal property of colimits, an important surjective map an injective map

\[ {\rm hom}(X,Y) \overset{\subseteq}{\longrightarrow} \prod_{g\in {\rm gen}(X)} {\rm hom}(\bar{\mathbb N}_{({\rm ord}(X),{\rm ht}_X(g))},Y) \times {\rm hom}(\bar{\mathbb N}_{({\rm ord}(X),\infty)},Y)^{{\rm ete}(X)}   \]

This is because the canonical epi map \(\bigsqcup_{i\in I}X_i\to{\rm colim}_{i\in I}X_i\) induces contravariantly a mono, thus injective, map \({\rm hom}({\rm colim}_{i\in I}X_i,Y)\to{\rm hom}(\bigsqcup_{i\in I}X_i,Y)\). Since hom-sets exponentiation sends sums to products and, in general, co-limits to limits, the map induced is the canonical one \({\rm lim}_{i\in I}{\rm hom}(X_i,Y) \overset{\subseteq}{\longrightarrow}\prod_{i\in I}{\rm hom}(X_i,Y) \).

Since we have that \({\rm hom}(\bar{\mathbb N}_{({\rm ord}(X),{\rm ht}_X(g))},Y)\) is in bijection with the sets of states of \(Y\) with height at most \( {\rm ht}_X(g)\), I call it \({\mathcal E}^{({\rm ord}(X),{\rm ht}_X(g))}(Y)\) because of some historical reasons related to the Grzegorczyk hierarchy, and since \({\rm hom}(\bar{\mathbb N}_{({\rm ord}(X),\infty)},Y)\) are just the set of eternal generators of \(Y\) we can simplify the notation as follows

\[ \boxed{ {\rm hom}(X,Y) \overset{\subseteq}{\longrightarrow} \prod_{g\in {\rm gen}(X)} {\mathcal E}^{({\rm ord}(X),{\rm ht}_X(g))}(Y) \times {\rm ete}(Y)^{{\rm ete}(X)} } \]

This map [has a section that] presents each superfunction \(X\to Y\) as a choice of heigts and eternal states. I'll add more about this decomposition and why matters imho.

note to self: this last formula requires some attention because in the case of order \(0\) and \(\infty\), i.e. aperiodic, the heigh is not easy to define coherently. Some more study of the correct definitions for the general case is still needed.

ADDENDUM 8/9/25: QUASI-MATRIX REPRESENTATION OF GENERAL SUPERFUNCTIONS
I uess we can combine both formulas and get the following injection for non-connected systems too:

\[ \boxed{ {\rm hom}(X,Y) \overset{\subseteq}{\longrightarrow} \prod_{\lambda\in \pi_0X}\bigsqcup_{\theta\in {\bf D}_{X,Y}(\lambda)} \prod_{g\in {\rm gen}(X_\lambda)} {\mathcal E}^{(o_X(\lambda), {\rm ht}_{X_\lambda}(g))}(Y_\theta) \times {\rm ete}(Y_\theta)^{{\rm ete}(X_\lambda)} } \]

I just oneed to check some trivial stuff... probably really easy facts about injections being preserved by products and sums. If this is true then this is as close as we can get to a matrix representation for arbitrary superfunctions. Not exatly matrices but vectors of vectors where each entry belongs to a completely separate function space, depending on the coordinate in the "matrix".

[Natsugou]

I don't know category theory nor colimit, but may I ask you questions about the definitions of \(\mathrm{gen}(X)\) and \(\mathrm{ete}(X)\)?
Is the definition of \(\mathrm{gen}(X)\) \(\mathrm{gen}(X) := \{x \in X \mid \forall y \in X, f(y) \neq x\}\)?
And which one of the three is the definition of \(\mathrm{ete}(X)\), where \(X' = \{x \in X \mid \exists! y \in X, f(y) = x\}\),
\[
  \begin{array}{ccccl}
    p: & X' & \to    & X' & \\
      & x  & \mapsto & y  & \mathrm{s.t.}\; f(y) = x,
  \end{array}
\]
\(X'' = \{x \in X' \mid \forall n \in \mathbb{N}, p^n(x) \in X'\}\), \((X'', p)\) is a \(\mathbb{N}\mathbf{Set}\),
and \(A\) is the set of decomposed connected dynamical sub-systems of \((X'', p)\)? (i.e. \((X'', p) = \bigsqcup_{(e, p) \in A}(e, p)\).)

1. \(\mathrm{ete}(X) := A\).
2. \(\mathrm{ete}(X) := \{(e, p) \in A \mid (e, p) \text{is generated}\}\).
3. \(\mathrm{ete}(X) := \{(e, p) \in A \mid (e, p) \text{is infinite}\}\).
           

(edit 2025-09-22) I noticed that my definition of \(\mathrm{ete}(X)\) written above doesn't work well when \(X\) is a dynamical system like the infinite complete binary tree. What is \(\mathrm{ete}(X)\) of such \(X\)?

[MphLee]

I did a bad abuse of notation. Also one of my claim was a little bit wrong because of this abuse.
I'll give you the wanted definitions and some good pictures, like the one you used, to aid our intuition.

i just need 24h.

---

[edit 23 semptember 1am] I'm sorry, I need more time for the pictures. In a couple of hours.

Btw \({\rm gen}(X)\) are the states without a preimage. The eternal generators \({\rm ete}(X)\) are equivalence classes of bi-infinite sequences of states that are "linked by \(f\) ". The generated part of \(X\) is generated by \({\rm gen}(X)\) , i.e. closed under application of \(f\) . The eternal part of \(X\) is made by the states that have at least one infinite chain of preimage.

Tonight I'll make a proper answer with the pictures.

[MphLee]

Imporoved 25/9/25 Fixes and typos

I did a bad abuse of notation there: I used the same notation for three different sets. The \({\rm ete}(X)\) in the exponent and the one in the base are two different sets. 

I'll start by drawing some pictures of four dynamical systems: \((\mathbb N, S)\),  \((\mathbb Z, S)\),  \((\mathbb N, h_{1,3})\) addition by 3,  \((\mathbb N, h_{2,3})\)multiplication by \(3\).


The first and second one have one connected component, the third has 3 connected components, the fourth has infinite components.
In general


COPRODUCT OF DYNAMICAL SYSTEMS
def 0. Obviously \(|\pi_0 (X\sqcup Y)|= |\pi_0 ( Y)| + |\pi_0 (Y)| \). Define a system \(X\) to be connected iff \(| \pi_0 (X) |=1\).

All the dynamical systems are sums of connected, simpler, dynamical systems.
Here a table of some of the simplest connected systems organized by order.


GENERATORS and GENERATED PART (from latin genus - birth, origin, lineage, descent )
def 1a) \({\rm gen}(X)=\{g\in X\,|\,\forall x\in X. f(x)\neq g \}\) this is NOT a subsystem of \(X\). It is just a subset of the set of states of the dynamical system.


def 1b) The "generated" part of \(X\) is a dynamical subsystem  \(X_{\rm gen}\subseteq X\) that is produced from the generators. \( X_{\rm gen} :=\langle {\rm gen}(X)  \rangle\). In other words  \( X_{\rm gen} :=\{x\in X\, |\,\exists n.\exists g\in  {\rm gen}(X) .\, f^n(g)=x\}\).


NON-GENERATED PART
Now, we say a system is generated iff \(X_{\rm gen}= X\). Otherwise \(X\) must have some kind of un-generated, "eternal", aspectin it.


ETERNAL PART
def 2) We define \(z\in X_{\rm ete}\) iff \(z\) has at least one infinite past, i.e. \(z\) has "infinite predecessors". Or \(X_{\rm ete}=\bigcap_{n=1}^\infty f^n(X)\).

def 2') (alternative phrasing). \(z\in X\)  is "eternal", \(z\in X_{\rm ete}\), if exists a sequence of states \(x_n\in X\) such that, for all \(n\in\mathbb N\), \(x_0=z\) and \(f(x_{n+1})=x_n\).

\(X_{\rm ete}\) can be seen as a dynamical subsystem of \(X\).

Observation 1: The set \({\rm hom}(\mathbb Z,X):=\{ \phi:\mathbb Z\to X \,|\, \phi(a+1)=f(\phi(a)) \}\) can be used to define \(X_{\rm ete}\). Notice that also \({\rm hom}(\mathbb Z,X)\) can be seen as a dynamical system: as a dynamical system we denote it as \([\bar{\mathbb Z},X]\), the notation comes from the internal hom-sets. The successor of \(\phi\) is \(f\circ \phi \).

Observation 2: We have a map of dynamical systems \({\rm ev}_0:{\rm hom}(\mathbb Z,X)\to X \). Sends \(\phi\) to \({\rm ev}_0(\phi)=\phi(0)\). It is indeed a map of dynamical systems because \({\rm ev}_0(f\circ \phi):=(f\circ \phi)(0)=f(\phi(0))\). This means that its image forms itself a dynamicals subsystem of \(X\).

def 2'') (image version). \( X_{\rm ete}={\rm im}({\rm ev}_0)\).


ETERNAL GENERATORS
Now, observe that \(X_{\rm ete}\) is a quotient of \({\rm hom}(\mathbb Z,X)\), i.e. we have a surjection \([\bar{\mathbb Z},X]\to X_{\rm ete}\).
But, as a dinamycal system "per se", \([\bar{\mathbb Z},X]\) can be quotiented naturally in connected components. These components are after the concept of what I call "eternal generator". Maybe we can find a better name for those, maybe στοιχεῖον could be adequate...

def 3) \({\rm ete}(X):=\pi_0 [\bar{\mathbb Z},X]\)


You can see how every system can be seen as an union (not a disjoint sum but a pushout) of its generated and its eternal part.

\[X= X_{\rm gen} \sqcup_{V_X} X_{\rm ete} \], where \(V_X\) is the place the two subsystem are glued, i.e. their intersection \(X_{\rm gen} \cap X_{\rm ete}=V_X \).

So a more precise formula should be something like

\[ \boxed{ {\rm hom}(X,Y) \overset{\subseteq}{\longrightarrow} \prod_{\lambda\in \pi_0X}\bigsqcup_{\theta\in {\bf D}_{X,Y}(\lambda)} \prod_{g\in {\rm gen}(X_\lambda)} {\mathcal E}^{(o_X(\lambda), {\rm ht}_{X_\lambda}(g))}(Y_\theta) \times {\rm hom}({\bar{ \mathbb Z}},Y_\theta)^{{\rm ete}(X_\lambda)} } \]

To be honest, some more fine tuning is needed in this formula, but I believe the essence of it is already there.

---

Addendum 25/09/25

Notice that the generated part and the eternal part are some kind of closure operators on the category \(\mathbb N{\rm Set}\).  Are idempotent operators.

\( ( X_{\rm ete} )_{\rm ete}=X_{\rm ete} \) and \( ( X_{\rm gen} )_{\rm gen}=X_{\rm gen}\)

also only only \( (-)_{\rm ete}  \) seems to be a functor.

[MphLee]

@Natsugou
I've started to notice many analogies on \( {\rm ete} (X)\) and on some other other contructions that "mods out"  by connectedness and the contruction of projective spaces.
I'm trying to make those analogies more precise and formal.

Connected components contruction is like giving the possible "direction" of rays (half lines) at positive infinity. Eternal generators is like modding out lines by the parallel direction, but in the direction of negative infinity. 
So both constructions are spiritually connected with the concept of projective space as the space of 1-dimensional subspaces (lines) of a vector space modded out by the parallel (scaling) relation.

In other words, if vector spaces over k have to do with the geometry of the field k, with dynamical system NSets have to do witht the "geometry" of the natural numbers (as a monoid).

Injective funtions/NSet are free like Nsets, like free vector spaces/modules over a Ring.
Surjective functions means that you can go back to negative infinity, so being surjective mean being "completed" by adding the eternal generators.

Much more is going on here, and much more have to do with tensor product of dynamical systems (induction/coinduction of representationa) aka change of base functors. Very exciting.