Is every diffeomorphism an element of a one parameter group of diffeomorphims?

I understand that a smooth vector field on a manifold $M$, generates a "flow"/one parameter group action, lets say $\sigma(t,s): \mathbb{R} \times M \rightarrow M$, and $\sigma_t: M \rightarrow M$ gives a one parameter group of diffeomorphisms. My question is, do every diffeomorphism have to be an element of a such group? My naive guess is no, but I am confused because I think set of diffeomorphisms also form "a" group. I would appreciate if you can give an example of a such diffeomorphism.

Solution 1:

No. Diffeomorphisms form a topological group $\text{Diff}(M)$ which in some respects behaves like an infinite-dimensional Lie group, but this topological group is not connected in general: it has a group $\pi_0(\text{Diff}(M))$ of connected components, the mapping class group of $M$, which is in general interesting. The only diffeomorphisms which can be part of one-parameter groups are those in the identity component.

For example, take $M = S^2$. There is a diffeomorphism $f : S^2 \to S^2$ given by taking the antipode, and because it acts by $-1$ on top homology $H_2(S^2)$, it cannot be in the identity component of $\text{Diff}(S^2)$.

Solution 2:

Complementing the answer of @QiaochuYuan, there exist diffeomorphisms $f : D^2 \to D^2$ which are in the identity component of $\pi_0(\text{Diff}(M))$ but which are not contained in any 1-parameter group of diffeomorphisms.

For an example of such an $f$, take three points $p,q,r \in \text{int}(D^2)$, and take any diffeomorphism $f$ preserving $\{p,q,r\}$ so that the restriction to $D^2 - \{p,q,r,\partial D^2\}$ has pseudo-Anosov isotopy class, preserving a geodesic lamination $\Lambda \subset D^2 - \{p,q,r,\partial D^2\}$ with respect to a complete finite area hyperbolic structure on $D^2 - \{p,q,r,\partial D^2\}$. One may construct $f$ so that $\Lambda$ is a minimal set on which $f$ acts with hyperbolic dynamics, including dense orbits, and so that the restriction of $f$ to $D^2 - \Lambda$ has wandering dynamics. It follows that anything which commutes with $f$ preserves $\Lambda$ as a set, and preserves the decomposition of $\Lambda$ into its 1-dimensional leaves. From this, it follows that $f$ is not part of a 1-parameter subgroup.

For a more conceptual understanding of this example, every surface diffeomorphism that is contained in a 1-parameter subgroup has zero topological entropy. And, every diffeomorphism which, on the complement of some finite set, is isotopic to a pseudo-Anosov diffeomorphism, has positive topological entropy.