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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.

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    $\begingroup$ 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. $\endgroup$ Oct 21, 2021 at 11:22

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The dynamics can change quite drastically. (There are more subtle things one can say about dynamics on contact manifolds; e.g. for closed contact 3-manifolds there are always at least two periodic Reeb orbits (and it is conjectured that there are either two or infinitely many orbits with no in between, which has been proven in the generic case). But I will not address these points.)

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.

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