Let $X$ be a Noetherian scheme. Let $f\colon X\rightarrow X$ be an integral self-morphism. If $x\in X$ is a closed point, I will write $\mathcal{F}_{1}^x$ for the coherent sheaf of $\mathcal{O}_X$-modules whose support equals $\{x\}$ and whose stalk at $x$ has length $1$ as an $\mathcal{O}_{X,x}$-module.
For a problem that I am working on, things will work nicely if I assume that $f$ satisfies the condition that follows:
For any closed point $x\in X$, if $y$ is in the $f$-orbit of $x$, then $$\forall n\in\mathbb{Z}_{\geq0}:\ \ \ \ \operatorname{length}_{\mathcal{O}_{X}}\big((f^n)^*\mathcal{F}^x_1\big)\leq \operatorname{length}_{\mathcal{O}_{X}}\big((f^n)^*\mathcal{F}^y_1\big).$$
I have two questions:
1) Is this condition equivalent or similar to any condition in classical dynamical systems, say over a smooth (real or complex) manifold?
2) Is this a reasonable assumption on $f$? In characteristic $p>0$ the Frobenius endomorphism trivially satisfies this condition, so the condition is not empty, but is there a reason that would make this condition impossible for other integral self-morphisms?
