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!