Question
A normed vector space or is a vector space over the real or complex numbers, on which a norm is defined.
While this definition does not mentions any metric, we know that the norm will induce a metric on the underlying set of the normed vector space. The additional metric is, so to speak, a property of a normed vector space, which can be infered from the definition without being part of the definition.
I am interested in such properties for dynamical systems that exist for every dynamical system without being part of the definition. What else can be said about dynamical systems in general?
Dynamical System Definition
One definition defines a dynamical system like this (I shortened it slightly):
You've got a set, and the points in it are moving around with the passage of time, and $\Phi$ is keeping track of the movement. $T$ as giving the time. $\Phi(0,x)=x$ says that at time zero the points are at their starting places. The third condition says that if you look at where the points are after $t_1$ seconds (or days, or centuries, whatever), and then you look at where they are $t_2$ seconds later, you find out where they are after $t_1+t_2$ seconds.
In my publication I am considering the general definition from wikipedia, where $T$ is not related to time in the physical sense:
A dynamical system is a tuple $(T, M, Φ)$ where $T$ is a monoid, written additively, $M$ is a non-empty set and $Φ$ is a function
$$\Phi :U\subseteq (T\times M)\to M$$
with
- $\mathrm {proj} _{2}(U)=M$ (where $\mathrm {proj} _{2}$ is the 2nd projection map)
- $I(x)=\{t\in T:(t,x)\in U\}$
- $\Phi (0,x)=x$
- $\Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1}, x)$ for $t_{1},\,t_{2}+t_{1}\in I(x)$ and $t_{2}\in I(\Phi (t_{1},x))$
The function $Φ(t,x)$ is called the evolution function of the dynamical system: it associates to every point in the set $M$ a unique image, depending on the variable $t$, called the evolution parameter. M is called phase space or state space, while the variable $x$ represents an initial state of the system.
Example: Local search
While I am interested in the general question above, here is an example from my research. There I am studying local search, which I define like this:
Given a target function $f(x), x\in X$, over a topological space $(X, \tau)$ and an objective regarding the target function (e.g. minimization), then local search optimization is defined as the family of functions $L = (l_t)_{t\in\mathcal T}$ with $t,T \in \mathbb N\cup\{ \infty \}, \mathcal T = \{t \mid 1 \le t \le T\}$, such that $x_{t+1}=l_t(x_t) \in N_t(x_t)$ induces a sequence $\{x_t\}_{t=1}^T$ (with $x_t \in X$) for neighborhood $N_t(x)$ over $X$ such that $f(x_{t+1})$ is closer to the objective than $f(x_t)$ (e.g. f(x_{t+1}) < f(x_t) in the case of minimization). The tuple $( \mathcal T ,X,L)$ is a semi-cascading discrete-time dynamical system.
Background (Why the interest?)
For my publication I defined $T, M, Φ$. Put together, they obviously make up a dynamical system. However, even though $T, M, Φ$ interlace, I just them, not defintion-independent properties of the dynamical system, the triplet ($T, M, Φ$). So, my supervisor said - because I am not using the dynamical system's properties - I should not mention in my publication that they make up a dynamical system. He said, it will make the read more difficult, since a reader might ask themselves, what the dynamical system (as a whole) is used for and whether they need to learn about dynamical systems if they want to understand the paper. I think he has a point and realized I do not know what - if any - properties dynamical systems have in general - except for the properties from the definition above.
This is not a dublicate to Properties of a dynamical system, because the other question is asking for a specific dynamical system.
