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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 $\mathbb S^2$)?

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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.

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  • $\begingroup$ This argument should work also for weakly mixing minimal homeomorphisms of the circle. Indeed, by a theorem of Furstenberg, weak mixing implies weak mixing of all orders. Thus $f^n(A)\times f^n(A)\times f^n(A)\times f^n(B)\times f^n(B)\times f^n(B) \cap A\times B\times C\times A\times B\times C \not= \emptyset$ for some $n \in \mathbb{Z}$, which is the same consequence of mixing used in YCor's argument. $\endgroup$
    – Veridian Dynamics
    Commented Jun 23, 2021 at 10:27
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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.

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  • $\begingroup$ I mentioned the first fact in a comment to mathoverflow.net/questions/253156 (more precisely $f^2$ has a fixed point! antipodal map has no fixed point). But this does not mean there's no topologically mixing self-homeomorphism of the 2-sphere (at least, there are topologically transitive self-homeomorphisms, which is not trivial). $\endgroup$
    – YCor
    Commented Oct 26, 2016 at 23:30
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    $\begingroup$ @YCor, if minimality is not required, then it does exist area-preserving diffeomorphisms of $\mathbb{S}^2$ which are in fact mixing (not just topologically mixing). Some examples can be found in the paper of Katok jstor.org/stable/1971237?seq=1#page_scan_tab_contents $\endgroup$
    – Alejandro
    Commented Mar 22, 2018 at 16:03

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