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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 $Q$?

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    $\begingroup$ You may be looking for the notion of Structural Stability: en.wikipedia.org/wiki/Structural_stability $\endgroup$
    – Lasse Rempe
    Aug 21, 2015 at 13:39
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    $\begingroup$ Of course, there are situations where you may not have structural stability, but still have "dynamical convergence" along certain sequence. For example, as pointed out below, the Julia set of a quadratic polynomial is not continuous at c=1/4. However, if you approach through the interval $c\in (0,1/4)$, then the Julia sets will converge (and the maps are even topologically conjugate on the Julia set - where they are topologically just the doubling map on the circle). $\endgroup$
    – Lasse Rempe
    Aug 21, 2015 at 13:42
  • $\begingroup$ This seems like the right concept, although I'm probably asking for something weaker than being 'locally topologically conjugate'. But this is a nice direction to look in. Thank you! $\endgroup$
    – Giraffro
    Aug 21, 2015 at 13:55
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    $\begingroup$ Another situation that comes to mind is the convergence of families of unicritical polynomials, parameterised as $z\mapsto (1+z/d)^d+c$, to the exponential family $z\mapsto e^z+c$. The convergence seems "dynamical" in a certain sense. There is an old preprint by Devaney-Goldberg-Hubbard, now published in two parts with additional co-authors. Also, additional results in a similar spirit by Kriete, Krauskopf and others; see arxiv.org/abs/0910.0743 by my former student Helena Mihaljević-Brandt. However, I am not sure this "dynamical convergence" has ever been fully formalised as a concept. $\endgroup$
    – Lasse Rempe
    Aug 24, 2015 at 11:11

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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 $1/4$.

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