I'm trying to find references to approximations of topological dynamical systems in the following sense:

A topological dynamical system $(X, f)$ consists of a topological space (typically compact Hausdorff, or even metrisable) and a continuous $f : X \to X$. We can topologise $C(X, X)$ in various ways, for instance with pointwise / compact-open / uniform topologies. Given such a topology $\tau$, we denote the topological space by $C_\tau(X, X)$.

Are there theorems along the lines of "Given a sequence $(f_n) \subseteq C_\tau(X, X)$, if $(f_n) \to f$ then the dynamical systems $(X, f_n)$ 'approximate' $(X, f)$"? Perhaps $f_n \in C_\tau(X_n, X_n)$ for some subspace $X_n$ and we can consider 'approximations' from subsystems? Or $\{f \in C_\tau(X, X) : (X, f) \text{ has property } P\} \subseteq \overline{\{f \in C_\tau(X, X) : (X, f) \text{ has property } Q\}}$ for dynamical properties $P$ and $Q$?