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?