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.

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 And also this post Lifting a Quadratic System to a non Vanishing vector field on $S^3$)

A polynomial vector field on the plane gives us an analytic vector field $X$ on $S^2$. 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 but it is not the case for higher degree polynomial vector field. 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.

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. 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" and difficulties occured in "Finitness theorem for Limit Cycles" are motivations for the first question of this post.



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