I would like to apply the known version of the conjectural formula (11) page !0 of the paper [Number theory and dynamical Lefschetz trace formula](https://arxiv.org/pdf/math/0204110.pdf).
**Disclaimer:** I do not have a complete understanding of this formula but I can get an sketch of it. I just know that both sides of the formula are not number but distribution. I also know the meaning of ingredient of the formula. I wish to apply it to find an appropriate application in limit cycle theory.
To start this way, I have three precise questions:
**Question 1:** In the right hand side of the formula (11), is not necessary to assume that there are only a finite number of non degenerate periodic orbits $\gamma$(As they are appeared in the sum $\sum$ of the right side of the formula)? Is it implicitly included in the assumptions of that formula? I think that even the non degeneracy assumption of periodic orbits does not **easily** imply the finite ness of such periodic orbits. Is not possible that a sequence of non degenerate periodic orbits accumulate to a non periodic orbit which is a kind of complicated and strange attractor?
**Question 2:**(This question is completely different from the previous one but there are [some motivations from this posts](https://mathoverflow.net/questions/332989/a-non-vanishing-vector-field-on-s3-whose-flow-does-not-preserve-any-transvers) And also this post [Lifting a Quadratic System to a non Vanishing vector
field on $S^3$](https://mathoverflow.net/questions/182139/lifting-a-quadratic-system-to-a-non-vanishing-vector-field-on-s3-or-t1?noredirect=1&lq=1))
A polynomial vector field on the plane [gives us an analytic vector field $X$ on $S^2$](https://mathoverflow.net/questions/178077/why-poincare-sphere-compactification-and-not-torus-compactification?noredirect=1&lq=1). Put $\tilde{X}$ for the obvious lifting of $X$ on $S^2\times S^1$ with $\tilde{X}=X+\partial/\partial{\theta}$
>Is there a quadratic polynomial vector field on $\mathbb{R}^2$ with Poincare compactification $X$ such that $\tilde{X}$ on $S^2\times S^1$ does not admit a 2 dimensional transversal foliation which is invariant under the flow of $\tilde{X}$.
**The reason we consider Quadratic system:**
Note that [every quadratic system is a geodesible vector field](https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273648#273648) but it is not the case [for higher degree polynomial vector field](https://mathoverflow.net/questions/273970/a-cubic-system-with-two-nested-limit-cycles-with-opposite-orientations?noredirect=1&lq=1). On the other hand, in dimension 2, if a vector field $X$ is geodesible then there is a transversal field $Y$ with $[X,Y] \parallel Y$ this implies that orbits of $Y$ are invariant under flow of $X$. So this make us to be hopeful a little that the product vector field $\tilde{X}=X+\partial/\partial \theta$ admit a transversal 2 dimensional foliation which is invariant under the flow of $\tilde{X}$. Existence of such transversal foliation is the key condition in the paper we linked in the first lines of this post.
**The reason we consider $S^2\times S^1$ rather than $S^3$:**
The lifting of the simplest vector field $X=0$ to the Hopf vector field on $S^3$ [does not admit a transversal foliation](https://mathoverflow.net/questions/332989/a-non-vanishing-vector-field-on-s3-whose-flow-does-not-preserve-any-transvers/333153#333153).
**Question 3:** When we lift a vector field $X$ to a non vanishing vector field $\tilde{X}$ on $S^3, S^2\times S^1\quad\text{or}\quad T^1 S^2$, it is possible that the preimage of a closed orbit, which is an invariant torus, would not contain any closed orbit, so we loose our closed orbits.[please see the comment by Sebastian Goette in this post](https://mathoverflow.net/questions/332979/irrational-closed-orbits-of-vector-fields-on-s2-limit-cycles-and-trace-formu). With terminologies in the linked paper, let we have a 3 manifold foliated by 2 dimensional leaves whch is compatible with a flow $X$. So is there an analogy of the formula (11) in page 10 of the linked paper in the first lines of this post whose right side depends on invariant torus of $X$ as well as closed orbits of $X$?
**Remark:** The concept ["Blue Sky catastrophe"](http://www.scholarpedia.org/article/Blue-sky_catastrophe#Historical_note) and difficulties occured in ["Finitness theorem for Limit Cycles"](https://bookstore.ams.org/mmono-94) are motivations for the first question of this post.