# Algebraic Dynamics over separated schemes

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)?

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For Q2, the answer is yes. If you don't like equalizers, you may take the scheme-theoretic intersection of the graph of $\phi$ with the diagonal. BCnrd gives a sheaf-theoretic definition of fixed locus in mathoverflow.net/questions/3190/… – S. Carnahan Dec 19 '11 at 2:22
Thanks! I do like equalizers! Just couldn't find this definition of fixed points in the literature, and was a bit surprised by that. – Mahdi Majidi-Zolbanin Dec 19 '11 at 2:58