[Question] Classifying dynamical system by connected components

[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".