TITLE: Approximation of topological dynamical systems?
QUESTION [7 upvotes]: 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 $Q$?

REPLY [5 votes]: In general, dynamical properties of dynamical systems are often quite unstable. The Mandelbrot set makes one nice parameter: the base equation $z \mapsto z^2 + c$ is varies continuously with respect to any reasonable topology on continuous functions. The Mandelbrot set contains large open regions (for instance, the central cardoid, or all the disks touching it). In the interior of these, the dynamics is stable: nearby maps are actually topologically conjugate. But outside of these almost all properties of the dynamics will depend quite sensitively on where you are. For instance, look up plots of Julia sets of the function $z \mapsto z^2 + c$ for $c$ close to $1/4$.