I  would  like to apply the  know  version of the  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  two  precise questions:

1)In the  left hand  side of the formula (11), is  it  necessary to assume  that there are only a  finite number of non degenerate periodic  orbits $\gamma$(which are appeared in the  sum $\sum$ of the left side of the 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?


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  $X$ to $S^2\times S^1$ 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.


As our third question, when we lift a  vector  field  $X$ to  a  non vanishing vector  field $\tilde{X}$, it is  possible that the preimage  of  a  closed orbit, which is an invariant torus, would not contain any closed orbit, so  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 left side depends on invariant torus rather than closed orbits?