Abstract: In this seminar we will briefly discuss some recent results on Hamiltonian dynamics and on conservative dynamics. Let $H$ be a Hamiltonian on a 4-dimensional symplectic manifold $M$, $e\in H(M)\subset\mathbb{R}$ and $\mathcal{E}_{H,e}$ a connected component of $H^{-1}({e})$ without singularities. Define a star system as a system having a neighborhood in which any system has any closed orbit and any singularity hyperbolic. The first result shows that a Hamiltonian star system, on a 4-dimensional symplectic manifold, is Anosov. Now, let $X$ be a divergence-free vector field defined on a closed, connected Riemannian manifold. We will discuss the equivalence between the following conditions:

• $X$ is a $C^1$-star vector field
• $X$ is in the $C^1$-interior of the set of expansive divergence-free vector fields.
• $X$ is in the $C^1$-interior of the set of divergence-free vector fields which satisfy the shadowing property.
• $X$ is in the $C^1$-interior of the set of divergence-free vector fields which satisfy the Lipschitz shadowing property.
• $X$ has no singularities and $X$ is Anosov.

To finish the seminar, we will discuss the generalization for Hamiltonian dynamics of a result proved by Bonatti and Crovisier for diffeomorphisms, whereby a $C^1$-generic conservative diffeomorphism is transitive.