To me it seems that is not always the case that a discrete dynamical system of the form
$$ x_{k+1} = f(x_k) $$
can be seen as the discretization of a continuous one like
$$ \dot{x}(t) = f(x), $$
while we can always go in the opposite direction.
My question is: is there some keyword I can search in the literature to understand better this connection or do you have any reference to suggest?
I would like to understand if there are some sufficient conditions guaranteeing the possibility of doing this transition, I don't even need a constructive way to build this continuous version, "just" existence would already be a great thing to understand.
To be clear, I would appreciate even results on this sort of "continuous prolongation" based on increasing the dimension of the phase space, i.e. for example for a discrete system of $\mathbb{R}^2$ if there is a dimension $d\geq 2$ such that on it there is a well defined continuous system whose discretization at certain time instants correspond to the discrete one. This question should be easier since there is not the constraint in the dimension.
I know it is quite a broad question, but I need at least some terminology to look for.
Up to now I have read about embeddability of homeomorphisms into flows and this seems to answer to my question in some cases, so I just wanted to know if the embeddability into flows is the right thing to study or if there are other perspectives.
P.S.
About getting an approximation of them I am finding very interesting this question Links between difference and differential equations? at the moment.
Best Answer
As you allude to in your question, there is indeed a construction which inputs a discrete dynamical system in the form of a self-diffeomorphism $f : M \to M$ of a smooth manifold $M$, and outputs a continuous dynamical system called the suspension flow of $f$. This suspension flow is defined on a manifold $M_f$ of one dimension higher than $f$, namely the quotient of $M \times \mathbb R$ under the equivalence relation generated by $(x,t) \sim (f(x),t-1)$. This suspension flow is also generated in the usual fashion by a vector field on $M_f$, namely the quotient vector field of the $\frac{d}{dt}$ vector field on $M \times \mathbb R$ equipped with coordinates $(x,t)$, $x \in M$, $t \in \mathbb R$. Thus, $f$ can be realized as the time $1$ map of the suspension flow, restricted to the image of $M$ embedded in $M_f$.
There are further generalizations of this construction beyond the realm of self-diffeomorphisms.
In the case of a self-homeomorphism $f : M \to M$ where $M$ is just a topological space, the construction of the suspension flow on $M_f$ works exactly as stated, producing an action of $\mathbb R$ on $M_f$. However, $M_f$ is not a smooth manifold and so its rather tricky to express the suspension flow as being generated by a vector field.
There is a still further generalization to any continuous self-map $f : M \to M$ where $M$ is again a general topological space. There is again a "suspension flow" construction, defined on the quotient of $M \times [0,\infty)$ by the equivalence relation $(x,t) \sim (f(x),t-1)$ for $x \in M$, $t \in [1,\infty)$. The reason for restricting to positive time is that $f$ itself can only be iterated with positive exponents; there is no inverse map $f^{-1}$.