Let's define a dynamical system as follow : A dynamical system is a triple $(T, X, \varphi) $ where T is a time set, X is a state space, and $\varphi : T \times X \rightarrow X $ is a continious function that fulfills these conditions for all $x \in X$ : \begin{equation} \varphi (0,x) = x \end{equation} \begin{equation} \forall s,t \in T : ~~\varphi(t+s,x) = \varphi (t, \varphi (s,x)) \end{equation}
There are many examples for dynamical systems on the metric spaces. I want to ask for some simple examples of dynamical systems on the other spaces such as Hilbert spaces, vector spaces, topological spaces or other possible spaces.
Any source or comment would be appreciated.
For any set, a transformation of the set $F : X \to X$ always gives rise to a discrete-time dynamical system.
For topological spaces: any continuous endomorphism (that is, maps the space back into itself) of the space gives rise to a continuous dynamical system.
For linear spaces: A square matrix $A$ gives rise to a dynamical system on $\mathbb{R}^n$ given by iterating $A$, and the asymptotic behavior of trajectories of this dynamical system can be read off of the eigendecomposition for $A$. Such dynamical systems serve as models of local behavior in smooth ergodic theory, i.e., the study of dynamical systems possessing derivatives.
For Hilbert spaces: you can do the same thing as for linear spaces, but there are also many interesting nonlinear examples, e.g., solution operators for PDEs acting on suitable spaces of functions.
Another common example is the flow map for a suitably nice ODE- this is where many examples of continuous time DS come from.
A good reference for many model examples of Dynamical systems is Brin & Stuck, Introduction to Dynamical Systems, or Guckenheimer/Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of vector fields. The latter reads more like an encyclopedia.