I'm looking for references about monotone dynamical systems (mostly in discrete time). I'm interested in problems of the following type:
Let $X$ be a topological or metric space with a partial order or preorder. Let $f:X\to X$ be monotone ($x \le y$ implies $f(x)\le f(y)$) and extensive ($x\le f(x)$). When does the sequence $f^n(x)$ converge topologically?
Are there any standard results? Where should I start?