Is there a compact metric space $X$ and a function $f:X\to X$ such that the dynamical system $(X, f)$ has the following three properties?
topologically mixing (a map $f$ is topologically mixing if for all nonempty open subsets $A$, $B$, there exists $N$ such that for every $n \geq N$ we have $f^n(A) \cap B \neq \emptyset$)
not positively expansive (a map $f$ is positively expansive if there exists an $c>0$ such that for all $x\neq y$ there exists an $n \geq 0$ such that $d(f^n(x), f^n(y))>c$)
Somewhat of relevance:
Since Jakobsen and Utz have shown that the circle does not admit an expansive homeomorphism, I was hoping there would be an example of a topologically mixing and minimal homeomorphism on the circle but YCor has shown that there is no such thing.
Hiraide has shown in particular that there exist no expansive homeomorphisms on the 2-dimensional sphere $\mathbb S^2$, so maybe there is some hope there.