# Examples of minimal dynamical systems on a non-compact space

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?

• Just take your favorite compact example and remove a $\phi$-orbit from it. – Moishe Kohan Mar 7 at 4:05
• 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). – Iian Smythe Mar 7 at 16:17
• I was commenting on the question in the title. I do not know about $N^N$. – Moishe Kohan Mar 7 at 16:25
• 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.) – Moishe Kohan Mar 7 at 16:56
• 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). – Alex Ravsky Apr 21 at 11:56