Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)? The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a homeomorphism nor is it minimal.
Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?
 A: By Lefschetz fixed point theorem, any continuous map $f\colon\mathbb{S}^2\to\mathbb{S}^2$ has a fixed point. So, there is no hope of getting any minimal homeomorphism on $\mathbb{S}^2$. 
However, there exist minimal mixing homeomorphisms on the $2$-torus. The first examples were constructed by A. Kochergin in http://iopscience.iop.org/article/10.1070/SM2002v193n03ABEH000636 and more recently Artur Avila has announced the existence of smooth minimal and mixing diffeomorphisms.
A: There's no topologically mixing self-homeomorphism of the circle. Indeed, pick 3 points, so that the complement of these 3 points consists of 3 intervals $A,B,C$. If $g$ is a self-homeomorphism such that $g(A)$ meets all of $A,B,C$, then it has to contain entirely one of $A,B,C$.
Therefore it is not possible that all the intersections $g(A)\cap A$, $g(A)\cap B$, $g(A)\cap C$, $g(B)\cap A$, $g(B)\cap B$, $g(B)\cap C$ be simultaneously nonempty. (If $f$ were a topologically mixing self-homeomorphism, $f^n$ for large $n$ would have to satisfy this property.)
On $S^2$ it's completely another question. 
