[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?