[Question] Classifying dynamical system by connected components

[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.