Smale's 12th Problem: Centralizers of Diffeomorphisms

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $M$ be a closed connected smooth manifold, where closed means compact without boundary. Fix a differentiability class $r\in\mathbb N\cup\{\infty\}$. Let $\operatorname{Diff}^r(M)$ denote the group of bijective $C^r$ maps $M\to M$ whose inverses are also $C^r$, with its $C^r$ topology. For $f\in\operatorname{Diff}^r(M)$ its centralizer in the same group is \[ Z^r(f)=\{g\in\operatorname{Diff}^r(M):g\circ f=f\circ g\}. \] It always contains $\langle f\rangle=\{f^n:n\in\mathbb Z\}$, using inverse iterates when $n<0$. Here "trivial centralizer" means $Z^r(f)=\langle f\rangle$, not that $Z^r(f)$ consists only of the identity. Put \[ \mathcal T^r(M)=\{f\in\operatorname{Diff}^r(M):Z^r(f)=\langle f\rangle\}. \] A subset of this function space is residual if it contains a countable intersection of open dense subsets; a dense $G_\delta$ is such an intersection. [S2]

Statement recorded in the supplied source.

Determine whether, for each $M$ and $r$ as above, the diffeomorphisms with trivial centralizer have the following properties:

1. $\mathcal T^r(M)$ is dense in $\operatorname{Diff}^r(M)$;
2. $\mathcal T^r(M)$ is residual in $\operatorname{Diff}^r(M)$;
3. $\mathcal T^r(M)$ contains an open dense subset of $\operatorname{Diff}^r(M)$.

These are the distinct genericity formulations of Smale's centralizer question, with the entire commuting group required to consist of iterates of $f$. [S2] For $r=1$, Source S2 proves the residual assertion and, for positive-dimensional manifolds, recalls a negative answer to the open-dense assertion; these facts do not decide every higher-regularity case. [S2]

Short English statement

Do diffeomorphisms whose only commuting diffeomorphisms are their own iterates form a dense set, a residual set, or a set containing an open dense subset, in each differentiability class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 64 (SMA-012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Bonatti, Sylvain Crovisier and Amie Wilkinson, The C1 Generic Diffeomorphism Has Trivial Centralizer, Publications Mathématiques de l’IHÉS 109 (2009), 185--244; arXiv:0804.1416, Section 1.1, Question 1 and the following C1 results, pages 3--4. https://arxiv.org/pdf/0804.1416.