# Example of overtwisted contact manifold with finitely many periodic Reeb orbits

Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits?

An example is given by the irrational ellipsoid in $$(\mathbb{R}^4,\omega_\text{st})$$, which is not overtwisted. The only other example that I know of are quotients of the last example on lens spaces. Actually, contact manifolds admitting only a finite number of periodic orbit are proven to "rarely exist" (https://arxiv.org/pdf/0809.5088.pdf).

(I am aware that under non-degeneracy condition on the contact form, the authors in https://arxiv.org/pdf/1701.02262.pdf prove that there are $$2$$ or $$\infty$$-many periodic Reeb orbits, but as far as I know nothing more is known if the contact structure is overtwisted).

• It is an open question for a 3-manifold: The conjecture is that a contact 3-manifold either has two (if it's a quotient of 3-sphere) or infinitely many embedded Reeb orbits. – Chris Gerig Jul 19 at 15:49
• And the introduction of Gardiner-Hutchings-Pomerleano's paper that you reference (Corollary 1.6 and footnote 3) shows that, assuming nondegeneracy, there are infinitely many on overtwisted Lens spaces (including 3-sphere). – Chris Gerig Jul 19 at 16:19