Cycles in algebraic dynamical systems

Question

Suppose you have an algebraic set $X \subset k^n$, where $k$ has finite characteristic, and an algebraic function $f: X \to X$.

• (A) Is there any cohomological, or more generally geometrical, obstruction to the dynamical system $f$ having finite cycles?
• (B) Is there a way to count them?

Explicitly, I am looking for solutions of the equation $f^k(x) = x$.

Here there are some variants which are harder, but more linked to my motivation:

1. Suppose $X$ is an (affine) ind-variety, i.e. we have $X= \bigcup_{j=1}^{\infty} X_j$ where $X_j \subset X_{j+1}$ are algebraic varieties of finite dimension over $k$, and $X_j \to X_{j+1}$ is a closed embedding. You can also suppose that $f$ is defined piecewise as $f_j : X_j \to X_{j+1}$ (but note the shift on the indices, so that you can't study the dynamical system on the singular strata). How answers to (A) and (B) change?
2. Is there any simplification if we suppose $k=\bar{k}$? In case of a generic field, can we deduce something by base changing to the algebraic closure?
3. Suppose $Y \subset X$ is an invariant subset under the action of $f$. Can we say something about cycles appearing in $X\setminus Y$?

I think that if something exists, it should be really cool. I also think to be partly deceived by the word "cycles" appearing both in cohomology and dynamical systems :) However, what makes me guess there could be something is that, in some cases, properties of dynamical systems are determined by the geometry of the underlying object, and I was wondering if 'having a cycle' in the context of algebraic geometry is such a property.

Of course, the best case scenario would be having a formula for the cycles of length $d$ in terms of the geometric action of $f$.

Edit. I also recalled that in the proof of Weil conjectures there was a use of the Lefschetz theorem, counting fixed points as an alternating sum of traces in $\ell$-adic cohomology. In my case $k$ is a finite field, so if this can be generalized to any map we could apply the Lefschetz trace formula to $f^k$ to get the desires result. Not being an expert though, I don't have any idea on how to formalize this, and if this is specific to the Frobenius map or to any map.