I [asked this in stackexchange](https://math.stackexchange.com/questions/1253347/is-there-a-diffeomorphism-with-only-finite-orbits-but-of-infinite-order), but got no answer, so I am trying here.


Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties: 


1. **All** its orbits are finite.
2. $\phi$ is **not** of finite order. (and hence has arbitrarily large finite orbits).


There are such examples if we allow infinitely many connected components (Take for instance countable disjoint copies of $\mathbb{S}^1$ and rotation of order $n$ in the $n$-th copy). 


So let us assume $M$ is connected. (I prove below that we can reduce the case of finitely many components to one component).


I will add that this question was inspired by a previous question of mine, about [global obstructions for a diffeomorphism to be an isometry](https://mathoverflow.net/questions/203330/is-there-a-global-obstruction-for-a-diffeomorphism-to-be-an-isometry) (You can see the **update** there for the relevant connection).




---


**Claim:** existence of a self-diffeomorphism for a manifold with finitely many connected components, implies existence for a connected manifold.


**Proof:** In this case the components $U\_i$ are clopen connected sets, hence $\phi(U\_i)$ are also clopen connected sets. Now combine the following two facts:


1. [clopen sets are always a union of connected components](https://math.stackexchange.com/questions/902313/why-are-clopen-sets-a-union-of-connected-components).
2. every connected subset of a topological space $X$ is contained in a connected component $X$.


Now it is evident that each $\phi(U\_i)$ is a connected component, i.e $\phi$ permutes the components. Since there are finitely many components, if we choose some component $U\_i$, we will get $\phi^n(U\_i)=U\_i$ for sufficiently large $n$. 


Next, we observe that the if $\phi$ has the two required properties, then $\phi^n$ also has them.


So finally, $\phi^n|\_{U\_i}$ is a self-diffeomoprhism of a **connected** manifold with the required properties.



is there a diffeomorphism with only finite orbits but of infinite order?
No, there is no such animal. This is due to Montgomery (1938), see my answer to this question: [Nonperiodic points of homeomorphisms of a ball](https://mathoverflow.net/questions/191133/nonperiodic-points-of-homeomorphisms-of-a-ball/191143#191143)


