Attractors as eventual epimorphisms

This is a short note about attractors in some dynamical systems.

An attractor is generally defined as a sort of “eventual approximate surjectivity”. Roughly speaking, a set is a global attractor if (almost) every point eventually gets near it. In other words, the inclusion A \hookrightarrow X is, in some sense, “eventually approximately surjective” — every point in X eventually gets near a point in A. How can we give a categorical formulation of this idea?

There are two ways to define “surjectivity” categorically: an inward way, and an outward way. The inward way — that is, the definition which concerns maps in — says that a function a : A \to X is surjective when it lifts on the right against the inclusion \emptyset \hookrightarrow \{\ast\}. This is an obvious sort of surjectivity, but it tends to be a bit “choice-y”; if we use a generalized elements formulation in an arbitrary category \mathcal{C} (for all Z, \mathcal{C}(Z, A) \to \mathcal{C}(Z, X) is surjective), then we find that “surjectivity” means having a section. This is too strong for most things we want to do.

The outward definition of surjectivity is a bit more mysterious: we say that a : A \to X is epimorphic when for every Z, the map \mathcal{C}(X, Z) \to \mathcal{C}(A, Z) given by precomposition is injective. Surjections in the category of sets are epimorphic, essentially by function extensionality, but it is a not entirely obvious fact that epimorphisms in the category of sets are surjections.

However, epimorphisms are not always surjections even in concrete categories whose objects are sets equipped with structure. There are two classic counterexamples:

In each of these cases, we see that while epimorphisms aren’t surjective, there is a sense in which their image controls the entirety of their codomain. There is an intuitive sense that a global attractor “controls” the entire dynamics of a system too, so it seems reasonable to use this kind of “surjectivity”.

The outward definition of “surjectivity” (that is, epimorphicity) relies on a prior notion of injectivity. So to weaken the notion of surjectivity to an eventual approximate surjectivity, what we actually need to do is weaken the notion of injectivity.

Definition: Let f_X : X \to X be a continuous endomorphism of a metric space (with possibly infinite distances). We say that x and x' are eventually equal if for every \varepsilon > 0 there is a N so that for any n > N, d_X(f_X^n(x), f_X^n(x')) < \varepsilon. We write this as x \approx x'.

We say that continuous morphisms of dynamical systems \phi, \psi : X \to Y are eventually equal when for every x \in X, \phi(x) \approx \psi(x). We write this as \phi \approx \psi.

Now we can give our definition of eventual epimorphism. We are working in the category of discrete time, continous dynamical systems on metric spaces (where the maps are asked only to be continuous, but not necessarily short).

Definition: A map \alpha : A \to X is eventually epimorphic if whenever \phi \circ \alpha = \psi \circ \alpha, we have \phi \approx \psi.

Suppose that A \hookrightarrow X is an invariant subset (that is, the inclusion is a morphism of systems) which is eventually epi. We can show that it is a global attractor in the following sense: for every x \in X and \varepsilon > 0, there is an N so that for n > N, we have that \inf_{a \in A} d_X(f_X^n(x), a) < \varepsilon. Consider the quotient map q : X \to X/A where X/A is the set (X - A) \sqcup \{A\}, equipped with the same metric as in X except that d_{X/A}(x, A) = \inf_{a \in A} d_X(x, a). Since A is invariant under f_X, f_X descends to X/A. Now q is constant on A by construction, so by the assumption the inclusion of A into X is eventually epi, we conclude that q is eventually equal to the constant map sending every x \in X to A \in X/A. This says precisely that A is an attractor.

I found going the other direction a bit more difficult. Let’s restrict our attention to the category of uniformly continuous maps. Suppose that A \hookrightarrow X is an invariant subset (that is, the inclusion is a morphism of systems) which is a global attractor in the above sense. Let \phi,\psi : X \to Y be uniformly continuous and suppose that \phi \circ \alpha = \psi \circ \alpha. Let x \in X and \varepsilon > 0, and let \delta(\varepsilon/2) be a modulus for the uniform continuity of both \phi and \psi. Let N be such that for n > N there is an a \in A such that d_X(f_X^n(x), a) < \delta; then d_Y(\psi(f_X^n(x)), \psi(a)) < \varepsilon / 2 and also d_Y(\phi(f_X^n(x)), \phi(a)) < \varepsilon / 2. By using that \phi(a) = \psi(a) and the triangle identity, we conclude that \phi \approx \psi.

Can this be improved upon (are these arguments even correct?)? The problem above is that if the continuity is not uniform, I need to know what N is before I know where to ask for my \delta. Perhaps if the dynamics are suitably bounded it wouldn’t be a problem.

What more can be said about eventual epimorphisms?


Not resolving anything, but the two thoughts that occurred to me are: (1) maybe uniform eventual equality is better-behaved than pointwise, and (2) it seems unnatural that eventual epimorphicity involves equality on the compositions and eventual inequality on the uncomposed maps. Did you write that specifically with an eye to making it work for attractors, or was there an organic motivation? It seems like if we strengthen that definition to \phi\circ\alpha\approx\psi\circ\alpha\implies\phi\approx\psi, we could talk about actual epis in the category of dynamical systems and eventual-equality-classes of maps, which might be fun.

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Yeah, you caught me; I did change it so that it would work for attractors. Uniform eventual equality might work better.

It also seems that for \approx to be a congruence, we need uniform continuity for the direction g \approx h \Rightarrow f \circ g \approx f \circ h. Or, rather than uniform continuity in general, we might ask for a “continuity uniform with respect to the dynamics”: For any x and \varepsilon > 0, there is a \delta such that for any x' \in X, d_X(f_X^n(x), f_X^n(x'))< \delta implies d_Y(\phi(f_X^n(x)), \phi(f_X^n(x'))) < \varepsilon. Basically, I’m not sure there is a category in which the eventual epis (weakened in the way you suggest) are the epis unless we restrict to uniformly continuous maps. Uniform eventual equality would probably resolve this, though. But then wouldn’t we end up with a uniformity on the notion of attractor as well?

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