TITLE: Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?
QUESTION [6 upvotes]: 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 $\mathbb S^2$)?

REPLY [4 votes]: 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.