# Change of Reeb orbits after scaling the contact form

Let $$M$$ be a contact manifold with a contact form $$\theta$$ with Reeb vector field $$X$$ and $$f$$ be a positive function on $$M$$. If $$\mathcal{L}_X f\neq 0$$ the Reeb vector field $$X'$$ of $$\theta'=f \theta$$ will not be a multiple of $$X$$. My question is: How much of the topology of the Reeb orbits changes? For example, can it happen that the number of closed Reeb orbits of $$X$$ and $$X'$$ are different? An example would be appreciated.

• To give you an idea of how much things change, just think that on the spherized cotangent bundle to a manifold $M$ (the bundle of cotangent directions), the contact forms associated to any two Riemannian (or even Finsler metrics) are such multiples. On the $2$-sphere you can go from all geodesics are closed to ergodic geodesic flow, to only two geodesics are closed. Oct 21, 2021 at 11:22

First, a helpful simplification. By Gray stability, you don't need to restrict to changes of the form $$\theta' = f\theta$$: if you study deformations $$\theta_t$$ (with $$\theta_0 = \theta$$) through contact forms, you can find a diffeotopy $$\phi_t$$ (with $$\phi_0 = id$$) so that $$\theta_t = f_t\phi_t^*\theta$$ for some family of positive functions $$f_t$$. So if the Reeb dynamics change drastically for such a deformation of $$\theta$$ (not necessarily of the form $$\theta_t = (tf+1)\theta$$ or something similar), then you have an answer to your question.
Example A standard example is the following (Geiges' An Introduction to Contact Topology Example 2.2.5): Consider on $$S^3 \subset \mathbb{R}^4$$ the contact form given by the restriction of $$\theta_t = (x_1dy_1 - y_1dx_1) + (1+t)(x_2dy_2 - y_2dx_2)$$ to $$S^3$$ for $$t > -1$$. Then the Reeb vector field is $$R_t = (x_1\partial_{y_1} - y_1\partial x_1) + \frac{1}{1+t}(x_2\partial_{y_2} - y_2\partial_{x_2})$$ which preserves the decomposition $$S^3 = \bigcup_{0 \leq C \leq 1} T_C$$ where $$T_C$$ are the tori $$T_C = \{|x_1|^2+|y_1|^2 = C, |x_2|^2+|y_2|^2 = 1-C\} \subset S^3.$$ (When $$C=0,1$$, we just have circles.) If $$t \in \mathbb{Q}$$, then on each $$T_C$$, we just have the flow by translation along a line of rational slope, and so we get that each $$T_C$$ is foliated by Reeb orbits. If $$t \notin \mathbb{Q}$$, then we are flowing with irrational slope, so there are only two Reeb orbits, the degenerate "tori" $$T_0$$ and $$T_1$$.
Same example, with appeal to Gray stability removed: If you want to translate this into the non-Gray-ified version, where you're just scaling $$\theta_0$$ on the nose, then notice that $$\psi_t \colon S^3 \rightarrow S^3$$ given by $$\psi_t(x_1,y_1;x_2,y_2) = \frac{(x_1,y_1,x_2\sqrt{1+t},y_2\sqrt{1+t})}{\sqrt{1+t(x_2^2+y_2^2)}}$$ satisfies $$\frac{1}{1+t(x_2^2+y_2^2)}\psi_t^*\theta_0 = \theta_t.$$ In other words, $$\psi_t$$ allows us to conflate the dynamics of $$\theta_t$$ with those of $$\frac{1}{1+t(x_2^2+y_2^2)}\theta_0$$.
Side Note: If you're curious about how this mysterious $$\psi_t$$ appears, the point is that these are constructed by considering the ellipsoids $$E_t = \left\{|x_1|^2+|y_1|^2 + \frac{|x_2|^2+|y_2|^2}{1+t}\right\} \subset \mathbb{R}^4.$$ We have that on $$\mathbb{R}^4$$, any star-shaped hypersurface $$H$$ is contact with respect to the restriction of $$\lambda = (x_1dy_1 - y_1dx_1) + (x_2dy_2 - y_2dx_2)$$, and the radial projection $$H \cong S^3$$ induces a contactomorphism. Hence, we have a radial contactomorphism $$\phi_1 \colon (S^3,\ker\lambda|_{S^3}) \xrightarrow{\sim} (\partial E_t,\ker \lambda|_{\partial E_t}),$$ i.e. with $$\phi_1^* (\lambda|_{\partial E_t}) = f_t\lambda|_{S^3} = f_t\theta_0$$. On the other hand, we have the isomorphism $$\phi_2 \colon S^3 \xrightarrow{\sim} \partial E_t$$ given by dilating the $$(x_2,y_2)$$-coordinates, and $$\theta_t$$ is just defined as $$\phi_2^* (\lambda|_{E_t})$$. Our $$\psi_t$$ is just $$\phi_1^{-1} \circ \phi_2$$.
Generic statement: Generically, you can make all Reeb orbits non-degenerate, which is a condition implying that no orbit has a nearby Reeb orbit of similar action. The point is that $$\theta' = e^f\theta$$ has Reeb field $$R_{\theta'} = \frac{1}{e^f}(R_{\theta} + X_f)$$ where $$X_f$$ is the unique vector field in $$\ker \theta$$ such that $$d\theta(X_f,Y) = df(Y)$$ for all $$Y \in \ker \theta$$, and so one can argue that any Reeb orbit can be made non-degenerate by choosing $$f$$ supported near that Reeb orbit appropriately. (C.f. Lemma 2 of Bourgeois' Introduction to Contact Homology.) In particular, the $$t\in \mathbb{Q}$$ case for our $$S^3$$ example is non-generic because none of the Reeb orbits are isolated. There are many many more examples of non-isolated Reeb orbits: prequantization/Boothby-Wang bundles, the standard contact form $$-\cos(z)dx+\sin(z)dy$$ on $$T^3 = S^1 \times S^1 \times S^1$$, etc. For all of them, a generic perturbation will kill this isolation.