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If $M$ is a connected $(d\geq 2)$-dimensional smooth closed manifold, then does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself, such that $(M,\phi)$ is a topologically mixing (discrete) dynamical system?

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  • $\begingroup$ True, I really only need a $C^1$-structure on the manifold. $\endgroup$ – N00ber Jul 19 at 8:22
  • $\begingroup$ This is answered (negatively) at this question for $d=1$. $\endgroup$ – YCor Jul 19 at 8:25
  • $\begingroup$ I forgot to put $d\geq 2$ (this seems possible since I know any compact manifold of dimension $d\geq 2$) admits a topologically transitive map. $\endgroup$ – N00ber Jul 19 at 8:34
  • $\begingroup$ OK. In a comment to another answer to that same linked question, it is said that the answer is yes for the 2-sphere (which can sound surprising). $\endgroup$ – YCor Jul 19 at 8:37
  • $\begingroup$ Ah, so there is hope that it is true in general... but is this the case then? $\endgroup$ – N00ber Jul 19 at 8:45
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Yes it's true.

It follows from a result of Abdenur and Crovisier (ArXiv link) [AC]: given a volume form $\omega$ on $M$ (closed manifold — the authors seem to omit assuming $\dim(M)\neq 1$), inside the topological group $\mathrm{Diff}^1(M,\omega)\subset\mathrm{Diff}^1(M)$, there is a Baire-generic subset of topologically mixing elements.

Note that it's not true in dimension 1 since then $\mathrm{Diff}^1(M,\omega)$ is reduced to the group of isometries (of some metric canonically determined by $\omega$, isometric to the standard circle), and actually there's no topologically mixing self-homeomorphism at all (see this answer).

Also, in the whole group $\mathrm{Diff}^1(M)$, the set of topologically mixing elements is not dense at all. Indeed, choose two small disjoint open subsets $U,V$ and $f$ with $f(\bar{U})\subset U$: then any close neighbor $g$ of $f$ (in the $C^0$, hence in the $C^1$ topology) satisfies $g(\bar{U})\subset U$ hence $g^n(U)\cap V$ empty for all $n\ge 0$, hence $g$ fails to be mixing.

[AC] F. Abdenur, S. Crovisier, Transitivity and topological mixing for C1 diffeomorphisms. Essays in mathematics and its applications, 1–16, Springer, Heidelberg, 2012.

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  • $\begingroup$ This is an amazing answer. Thanks YCor :) $\endgroup$ – N00ber Jul 21 at 12:23

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