# Existence of topologically mixing (discrete) dynamical system on manifold

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?

• True, I really only need a $C^1$-structure on the manifold. – N00ber Jul 19 at 8:22
• This is answered (negatively) at this question for $d=1$. – YCor Jul 19 at 8:25
• 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. – N00ber Jul 19 at 8:34
• 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). – YCor Jul 19 at 8:37
• Ah, so there is hope that it is true in general... but is this the case then? – N00ber Jul 19 at 8:45

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.