When is one dynamical system an approximation of another?

Question

I've been thinking about the question of when a discrete time dynamical system $f : X \to X$ (or possibly other objects) can be said to approximately model another dynamical system. So far I've mostly failed to find this concept in textbooks or papers. I'm sure I'm not the first person to think of these, so I'll write down the definitions I've thought of, and I'd love it if someone could point me to relevant references!

The first two concepts are ones I have found mentions of, but they're both what I would call "exact" or "crisp" models, rather than approximate. I'm writing them out for clear reference and to contrast with.

Isomorphic

The most obvious one is that a system $g : Y \to Y$ is isomorphic to $f : X \to X$ if there exists a bijective mapping $\phi : X \to Y$ such that $g \circ \phi = \phi \circ f$.

This is a good starting point, but it's not exactly an "approximation".

Homomorphic

We could say that a system $g : Y \to Y$ is homomorphic to $f : X \to X$ if there exists a (not necessarily bijective) mapping $\phi : X \to Y$ such that $g \circ \phi = \phi \circ f$.

This is the same thing, except where the system $(X, f)$ gets "course-grained" into a partition that is then isomorphic to $(Y, g)$.

This is in some sense more of an approximate model, since $(Y, g)$ throws away some information about $(X, f)$. But it's still "exact" in the sense that the partitions must be obeyed exactly.

Since $(X, f)$ is a dynamical system, we want to iterate $f$ repeatedly. That means a non-trivial partition will only work if $(X, f)$ was decomposible in the first place. This means that ergodic systems have no non-trivial course-graining.

Measure approximation

This one makes sense after we put a probability measure $\mu$ on $X$. Let's say a system $(Y, g)$ is a measure-$\epsilon$ approximation of $(X, f)$ if it is isomorphic over a subset $E \subset X$ where the measure of $E$ is $1-\epsilon$.

An interesting case here would be a measure-0 approximation, which is, of course, different from an isomorphism.

It also seems natural to combine this with the above, where we have a measure-$\epsilon$ homomorphism.

This seems related to thermodynamics. Macrostates (such as temperature) are like partitions, and we can say that within probability $\epsilon$, a given microstate in that macrostate will evolve into a microstate in some other given macrostate.

Metric topology approximation

And this one makes sense after we define a metric $d(y_1, y_2)$ on $Y$. Let's say a system $(Y, g)$ is a metric-$\delta$ approximation of $(X, f)$ if the isomorphism holds to within $\delta$. That is, for all $x \in X$,

$$d( g \circ \phi, \phi \circ f ) < \delta$$

The two types of approximation above are totally different. If you have a system with both a measure and a metric, then you could have a mapping $\phi$ that is not isomorphic for any $x$, but which is metric-0. And you could have a mapping which is measure-0 isomorphic, but where the supremum over that measure-0 set of the distance is infinite.

Algorithmic information theory

Here's a final one, which is actually the first one I thought of, because it's closest to my research interests. I'm assuming this one is not in the literature.

Let's say that we have a set of "descriptions", which are essentially finite binary strings. We also have a mapping from said descriptions to a set of dynamical systems. A system $g$ is an $n$-bit approximation of $f$ if there exists a binary string $b$ of length $n$ such that concatenating $b$ with the string for $g$ gets mapped to a system that is isomorphic to $f$. (What I'm really thinking of here is that the strings are inputs to a prefix-free, additively optimimal universal Turing machine, such that the dynamical systems are computable functions.)

So, just to reiterate my question, is any of this in the literature?

Answer 1

The concepts defined in 1. and 2. are well-known and called just "conjugacy" and "semiconjugacy". But more relevant, if you have a topology, is the standard concept of "topological conjugacy", which is basically what you call "isomorphism" with $\phi$ a homeomorphism. This concept can be particularized to differentiable conjugacy by requiring that $\phi$ is a diffeomorphism if there is enough structure to allow differentiation. Topological conjugacy is crucial for the study of structural stability of systems.

This may be too strong, I guess, for your needs. But topological conjugacy has been weakened in various ways. You can take a look at this old paper and this recent paper, for instance.

Another quite obvious direction is that of introducing a a metric in a certain collection of dynamical systems, so that you can directly "measure" their distance. There are various very well-known ways to do so, depending on the collection of systems you're interested in. I suggest this very nice book.

As for 3., I didn't see exactly the same concept in the literature, but if you replace "measure $1-\epsilon$" by "co-meagre set of points", and take $\phi$ a homeomorphism, you have a known weakening of topological conjugacy (sometimes called "almost-topological conjugacy").

About 4., I think the concept is a bit strange, because you're only looking at what happens in $Y$. An extreme case to show the point: if you take $\phi(x)= y$ for every $x\in X$, with $y$ a fixed point for $g$, then $X$ and $Y$ are "metric-0 isomorphic" no matter what $(X, f)$ (and even $(Y,g)$ outside $y$) do.

Finally, concerning point 5., it is possible that your idea is related to symbolic dynamics, as suggested by Qiaochu Yuan, but I'm not sure. I don't think I really understand what you mean here, but in symbolic dynamics one "approximates" a dynamical system by starting (in the simplest possible setting) with what’s basically a partition of the phase space and then assigning a symbol to each partition element. Here you talk about maps from finite strings to systems, not maps from spaces to an alphabet of symbols. Maybe something more useful can be said if you clarify a little bit.