Miguel Abreu

(work with Macarini)

Problem: existence of elliptic orbits (i.e. all eigenvalues of return map on unit circle)

Elliptic closed orbits give many dynamical consequences (e.g. KAM tori; transverse homoclinic connections?; positivity of topological entropy)

Conjecture of Ekeland: every convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit.

In general, still open.

Thm (Dall Antionio, D'Oriofrio & Ekeland '95)
If $M \subset \mathbb R^{2n}$ convex and invariant by antipodal map, then carries an elliptic closed orbit.

Goal: understand and generalize this using contact homology.

Thm (Abreu-Macarini)

Consider a prequantization manifold $(M, \xi = \ker \beta)$ where $\beta$ generates a free $S^1$ action.

Let $\alpha$ be some other contact form. Let $G \subset S^1$ be a finite subgroup. Suppose $\alpha$ is invariant under the $G$ action.

Let $a$ be the free homotopy class of the fibre of $M$.

Assume one of the following holds:

• $M/S^1$ admits a Morse function with only even indices
• $a^j \ne 0$ for all $j \ge 1$.

Then for every $G$-invariant positively [negatively] $a$-dynamically convex form $\alpha$ has an elliptic orbit representing the class $a$ with $CZ(\gamma) = k_-$ [respectively $k_+$].

Definition: Fix a free homotopy class of loops $a$. Associate to $a$ the minimal degree $k_-$ and maximal degree $k_+$ in which contact homology is non-zero for this class.

We say positively $a$-dynamically convex if all closed orbits in the free class $a$ have at least CZ index $k_-$. (Negatively, at most $k_+$) [ok, modulo the grading shift or whatever]

Applications:
Suppose have metric $g$ on $S^2$ with $\frac{1}{4} \le \kappa \le 1$. Then there is an elliptic closed geodesic.

Proof: Harris-Paternain proved that this curvature condition means the geodesic flow lifts to dynamically convex on $S^3$ + perturbation.

Rmk different proof exists (Ballmann, Thorbergsson & Ziller) and furthermore maybe better result due to Contrera and Oliveira (http://arxiv.org/abs/math/0312005)

Also an application to magnetic flows. $T^*N$ with $\omega_0 + \pi^* \Omega$ for a closed 2-form $\Omega$. Take Hamiltonian from geodesic flow.

Benedetti: If $N$ is a surface of genus $g \ne 1$, and $\Omega$ is symplectic on $N$, then small energy levels are contact-type. If $g=0$, lifts to $S^3$ and is dynamically convex.

If $g > 1$, there exists a $|\chi_N|$ covering $\tilde M \to H^{-1}(c)$ such that $\tilde M$ is prequant
and the lift to $\tilde M$ is negatively dynamically convex.

Toric contact manifolds:
$(M^{2n-1}, \xi)$ has $\mathbb T^{n}$ action. (equivalently: symplectization is toric symplectic cone)

"Good" toric contact manifolds: $\mathbb{C}^d // K$ for some $K \subset \mathbb T^d$ linear sub-torus.

Miguel made an interesting remark about being able to construct examples in higher dimensions with finitely many periodic orbits. This is related to a question from Michael's talk. He wondered then if the correct generalization of Michael's conjecture is to suggest that the only manifolds with finitely many distinct orbits are these toric contact manifolds.