Let $M$ be a simple hyperkähler manifold, that is, a simply connected compact holomorphically symplectic manifold of Kähler type with $h^{2,0}=1$. Assuming $b_2(M)\neq 5$, we prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kähler cone with finitely many orbits. This statement is known as Morrison-Kawamata cone conjecture for hyperkähler manifolds. As an implication, we show that a hyperkähler manifold has only finitely many non-equivalent birational models. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian manifold of dimension at least three, constant negative curvature and finite volume, and $\{S_i\}$ an infinite set of complete, locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.