For this question, let's say that a dynamical system is a pair $(X,\phi$) where $X$ is a Hausdorff topological space and $\phi:X\to X$ a homeomorphism. Equivalently, one can think about continuous actions of $\mathbb{Z}$ on $X$. In this setting, $X$ is called the phase space of the system.

$(X,\phi)$ is minimal if it has no non-empty, proper, closed subsets which are invariant under $\phi$ and $\phi^{-1}$.

Typically, one studies compact phase spaces. For instance, there are many concrete examples of minimal systems on $X=2^\mathbb{N}$, the Cantor space, e.g., Toeplitz shifts, odometers, etc.

I am wondering about the situation when $X$ is not (locally) compact, particularly on the Baire space $X=\mathbb{N}^\mathbb{N}$ (with the product topology, where $\mathbb{N}$ is discrete). Here is my question:

Are there minimal dynamical systems with phase space $\mathbb{N}^\mathbb{N}$? What about concrete examples and references on this topic?

  • 1
    $\begingroup$ Just take your favorite compact example and remove a $\phi$-orbit from it. $\endgroup$ – Moishe Kohan Mar 7 at 4:05
  • $\begingroup$ Moishe, why is this always homeomorphic to $\mathbb{N}^\mathbb{N}$? I suppose you just need to show that it is nowhere compact (or something close to this). $\endgroup$ – Iian Smythe Mar 7 at 16:17
  • $\begingroup$ I was commenting on the question in the title. I do not know about $N^N$. $\endgroup$ – Moishe Kohan Mar 7 at 16:25
  • 1
    $\begingroup$ One suggestion is to consider the Bernoulli shift on $Z^Z$. This is not minimal, of course, but look for a suitable ergodic measure of full support. Almost every orbit then will be dense. This gives you an interesting minimal subset. Se if you can prove that such a subset is homeomorphic to $Z^Z$. (For this, you may have to impose further conditions on the measure.) $\endgroup$ – Moishe Kohan Mar 7 at 16:56
  • $\begingroup$ I recall that by Alexandrov-Urysohn Theorem every Polish zero-dimensional space for which all compact subsets have the empty interior is homeomorphic to the space $\mathbb N^{\mathbb N}$, see, for instance, Theorem 7.7 (here and here) in ”Classical Descriptive Set Theory” by A. Kechris (Springer, 1995). $\endgroup$ – Alex Ravsky Apr 21 at 11:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.