Contactomorphisms have in general no fixed points

Question

Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other words, there exists $g : V \to \mathbb{R}$ such that $$ \phi^* \alpha = e^g \alpha. $$ In symplectic topology, a major driving force is the Arnold conjecture, which relates the number of fixed points of a Hamiltonian symplectomorphims to topological invariants of the symplectic manifold (depending on the version, the invariant can be the minimal number of critical points of a function, the sum of the Betti numbers, or the cuplength). Of course, the surprise here is that in general, symplectomorphisms might not have fixed points (for instance a rotation or the torus endowed with its standard symplectic form). However, as soon as they are Hamiltonian, fixed points appear.

In the contact setting, the analogue of the Arnold conjecture was introduced by S. Sandon, and pertains to a different kind of points (for contactomorphisms), called translated points. From what I understand, Sandon chooses to deal with translated points instead of fixed points because contactomorphisms need not have any fixed points, even if they are obtained as time-$1$ map of contact isotopies. This apparently comes from the odd-dimensionality of the contact manifold $(V, \xi)$, but I don't understand why. Since the Euler characteristic of an odd-dimensional manifold is $0$, the Lefschetz fixed point theorem implies that for any diffeomorphism of $V$ which is isotopic to the identity, the sum of the indices of its fixed points must be $0$, but the indices might cancel each other out, so this does not imply that there as no fixed points at all.

Is there a reason why odd-dimensionality prevents "most" transformations to have fixed points?

Thanks a lot in advance!

Answer 1

I hope that the following answers some parts of the question.

1) a) It is not true that a generic contactomorphim doesn't have fixed points. For example, let $M$ be the three-dimensional torus that is the projectivisation of the tangent bundle to the two-torus $\mathbb T^2=\mathbb R^2/\mathbb Z^2$. Any self-diffeomorphism of $\mathbb T^2$ induces a concatomorphsm of $T^3$. Consider an Anosov sel-diffeomorphism of $\mathbb T^2$, for example, given by the Arnol'd cats map $$(x,y)\to (2x+y, x+y)\; {\rm mod }\;1$$ https://en.wikipedia.org/wiki/Arnold%27s_cat_map

This map has a unique fixed point on $\mathbb T^2$ and it has two fixed points on the projectivisation of the tangent bundle $T^3$. If one calculates the eigenvalues of this map of $T^3$, one sees that they both have real eigenvalues of modulus different from $\pm 1$. So if we slightly perturb the contactomorphism it will still have two fixed points.

b) If you want a similar example given by a 1-parameter family, one can take some self-diffeo of $\mathbb S^2$, induced by a linear map of $\mathbb R^3$, $(x,y,z)\to (e^{at}x, e^{bt}y, e^{ct}z)$, and again take the induced map on the projectivised cotangent bundle of $\mathbb S^2$.

2) Even if we think about symplectic geometry, the negation of always having fixed points of a symplectomorphism, is not "never having fixed points". For example if we look at the last geometric theorem of Poincare: https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff_theorem which was important for the whole development of symplectic geometry, we see that one can not drop the condition on the annuls self-map to be area preserving. If we drop this condition we can have both generic maps that have fixed points and generic maps that don't have fixed points.

3) Finally, informally, I would guess, that though the circle $S^1$ is usually not considered as a contact manifold, it has some of the features. One should probably think that the group of contactomorphisms of $S^1$ is the group of its diffeomorphisms. It is not true that generic diffeomorphsims of $S^1$ don't have fixed points.

Answer 2

Here's a complement to Dmitri's excellent answer. While it's true that contactomorphisms can have fixed points, even for some open set of contactomorphisms, there is a reason you don't expect to be able to detect these using something like Floer theory: if you could, it would tell you there are fixed points for all contactomorphisms, but it's very easy to construct contactomorphisms (even $C^{\infty}$-close to the identity) that don't have fixed points. Pick a contact form $\alpha$ and let $R$ be its Reeb field. This is nowhere vanishing by definition (since $\alpha(R)=1$). Flow along $R$ for very small time. This is a contactomorphism with no fixed points.

The difference between symplectic and contact is the following. While a symplectic manifold is like the full phase space of a dynamical system, a contact manifold is like the level set of a Hamiltonian. The Hamiltonian will usually have critical points (fixed points of the flow), but if you pick a single level set you're unlikely to pick a critical level set, so you're unlikely to see any fixed points.