On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

We study the global topology of the space $\mathcal{L}$ of loops of contactomorphisms of a non-orderable closed contact manifold $(M^{2n+1},\alpha )$. We filter $\mathcal{L}$ by a quantitative measure of the ``positivity’’ of the loops and describe the topology of $\mathcal{L}$ in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of $\mathcal{L}$ are subgroups of the homotopy groups of the subspace of positive loops ${\mathcal{L}}^{+}$. We obtain analogous results for the space of loops of Legendrian submanifolds in $(M^{2n+1},\alpha )$.