I have a few questions regarding the current status of research on algebraic dynamics over separated schemes. In what follows $\varphi:X\rightarrow X$ will be a finite self-morphism of a noetherian separated scheme. By looking at literature it appears to me that in most of the work done in the field of algebraic dynamics $X$ is assumed to be a variety over a field, often a projective variety over a field. This makes sense because algebraic dynamics is motivated by its arithmetical applications.
Q1 Have dynamical systems $(X,\varphi)$ been studied when $X$ is NOT a variety over a field (in particular, not a projective variety over a field)?
When $X$ is a projective variety over an algebraically closed field $k$, the scheme of fixed points of $\varphi$ is defined as the set of points in $X(k)$ that are fixed by $\varphi$.
Q2 Is there a definition of the scheme of fixed points of $\varphi$ when $X$ is NOT a variety over a field (in particular, not a projective variety over a field)? It appears that it must be defined as the equalizer of $\varphi$ and identity, but I cannot find this in the literature.
Q3 Assuming the answer to Q1 is yes, are there any known results about number of fixed points of $\varphi$ (e.g. finiteness results) when $X$ is NOT a variety over a field (in particular, not a projective variety over a field)?
