Is it possible for topological dynamical systems to be semiconjugate if one of the dynamical systems is minimal but the other one is not?

Question

Is it possible for topological dynamical systems to be semiconjugate if one of the dynamical systems is minimal but the other one is not? (minimal means every orbit of the dynamical system is dense) If so, is there any simple example of this?

Answer 1

Yes, it is certainly possible. One of the best known examples is the Denjoy construction. Full details of this construction are given in Milnor's notes, section 15B, under the title "Denjoy counterexamples". Let me give a brief outline.

Start with an irrational rotation of the circle $f : S^1 \to S^1$, $f(z) = e^{2\pi i \theta}z$ where $\theta$ is irrational. This map $f$ is minimal.

Pick an orbit, say $A = \{z_n \mid n \in \mathbb Z\}$ where $z_n = e^{2 \pi i \theta n}$. Pick a bi-infinite convergent sequence of reals $(\ell_n)_{n=-\infty}^\infty$, $\sum_{n=-\infty}^\infty \ell_n = 1$.

You then cut open $S^1$ at each point $e^{2\pi i \theta n} \in A$, and insert an open interval $I_n$ of length $\ell_n$ between the resulting two points (Milnor explains what happens with those two points carefully). The resulting space $X$, with the obvious circular ordering, is homeomorphic to the circle. The union of the open intervals $I_n$ is an open subset $W \subset X$, and its complement $X-W$ is a Cantor set.

You then define $\tilde f : X \to X$ to be the homeomorphism that takes each $I_n$ to $I_{n+1}$ by the obvious similarity map of expansion factor $\ell_{n+1} / \ell_n$. This map $\tilde f$ is not minimal, because the union of the $I_n$'s is a wandering set. Also, the map $X \mapsto S^1$ which collapses (the closure of) each $I_n$ back to the point $z_n$ is a semiconjugacy from $\tilde f$ to $f$.

As a side note, the map $\tilde f$ takes the Cantor set $X-W$ to itself, and the restricted map $\tilde f : X-W \to X-W$ is minimal.