4
$\begingroup$

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 $\mathbb S^2$)?

$\endgroup$
9
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 Oct 26 '16 at 23:30
  • $\begingroup$ @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 Mar 22 '18 at 16:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.