For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$.

For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, is the set of limit points of the orbit of $x$. That is, $$\omega(x) = \bigcap_{n \in \omega} \overline{\{f^m(x) : m > n\}}.$$ It's easy to check that $(\omega(x),f)$ is a dynamical system (i.e., $\omega(x)$ is closed under $f$ and closed topologically). Let's say that $(X,f)$ is an abstract $\omega$-limit set if it is isomorphic to the $\omega$-limit set of some point in some dynamical system.

Is there a dynamical system $(X,f)$ such that every abstract $\omega$-limit set is a quotient of $(X,f)$?

[In this context, an isomorphism $(X,f) \rightarrow (Y,g)$ means a homeomorphism $h: X \rightarrow Y$ such that $h \circ f = g \circ h$. A quotient $(X,f) \rightarrow (Y,g)$ means a continuous surjection $h: X \rightarrow Y$ such that $h \circ f = g \circ h$.]

I suspect the answer might be negative, but I can't think of a way to get at a proof.

Motivation: If we replace "compact metric" with "compact Hausdorff" (to broaden our notion of dynamical systems a bit), then there is a universal abstract $\omega$-limit set, namely $\mathbb{N}^* = \beta \mathbb{N} \setminus \mathbb{N}$ together with the shift map (the "shift map" being the map sending an ultrafilter $\mathcal F$ to the ultrafilter generated by $\{A+1 : A \in \mathcal F\}$). It seems to me that some smaller system should suffice to "capture" just the metric systems. Ideally, one would like a metric system to do this -- hence the question!

Potentially helpful information: The metrizable abstract $\omega$-limit sets have a very nice topological characterization due to Bowen. Namely, a metrizable system $(X,f)$ is an abstract $\omega$-limit set if and only if it is chain transitive. [$(X,f)$ is chain transitive if for some (equivalently, any) metric $d$ on $X$, for any two points $x,y \in X$, and for any $\varepsilon > 0$, there is a sequence $x = x_0, x_1, x_2, \dots, x_n = y$ such that $d(f(x_i),x_{i+1}) < \varepsilon$ for all $i < n$. The idea here is that this sequence is a "pseudo-orbit" (with error $\varepsilon$) from $x$ to $y$.]

Also possibly relevant is the well-known topological fact that every compact metric space is a continuous image of the Cantor space. So if $(X,f)$ is a system providing a positive answer to my question, I suspect $X$ should be the Cantor space.

  • $\begingroup$ Do you know the answer for the weaker cases? For example, is there even a system which is metrizable and universal for minimal systems in the sense that every minimal system is a quotient? $\endgroup$
    – Burak
    Jun 18, 2015 at 17:22
  • $\begingroup$ @Burak: I do not. I will point out that in the non-metrizable setting, there you can find a universal minimal system that is itself a minimal system (any minimal subsystem of $\mathbb{N}^*$ will do). So again the situation is very nice for compact Hausdorff spaces, but I don't know the answer for metric spaces. $\endgroup$
    – Will Brian
    Jun 18, 2015 at 17:28

1 Answer 1


The answer is no, there is no universal $\omega$-limit set.
The same is true for the classes of metric minimal dynamical systems and metric dynamical systems in general. I have written up proofs of these facts:


The basic idea of the proof is as follows: as Will Brian already conjectured, it is enough to consider dynamical systems with 0-dimensional phase spaces. Now, if $(X,f)$ is a dynamical system with a 0-dimensional metric phase space $X$ and there is a morphism from $(X,f)$ onto a dynamical system $(Y,g)$ with $Y$ 0-dimensional, then the Boolean algebra of clopen subsets of $Y$ embeds into the algebra of clopen subsets of $X$ by an embedding that respects the dynamical systems. Since $X$ is metric, its Boolean algebra of clopen subsets is countable. Using Sturmian subshifts we can cook up enough metric minimal dynamical systems $(Y,g)$ (that are automatically abstract $\omega$-limit sets) so that not all the corresponding Boolean algebras (with actions on them) embed into a single countable Boolean algebra (with action), showing that there are no metric universal dynamical systems.

The most general result is this: There is no metric dynamical system that maps onto all Sturmian subshifts. The Sturmian subshifts are minimal and hence abstract $\omega$-limit sets. Here it doesn't make a difference whether we consider $\mathbb Z$-actions, i.e., compact spaces with a homeomorphism, or $\mathbb N$-actions, i.e., compact spaces with a continuous map as in the original question.

  • $\begingroup$ This is a great answer: your proofs are interesting and clear, and you've thoroughly answered my question. Sorry it took me so long to respond, but I wanted to read what you'd written first. $\endgroup$
    – Will Brian
    Aug 17, 2015 at 21:50
  • $\begingroup$ Thanks a lot for your kind comment and for taking the time to really look at the proofs! $\endgroup$ Aug 18, 2015 at 6:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.