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

We study the global topology of the space L of loops of contactomorphisms of a non-orderable closed contact manifold (M²ⁿ⁺¹, α). We filter L by a quantitative measure of the ``positivity'' of the loops and describe the topology of L in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of L are subgroups of the homotopy groups of the subspace of positive loops L⁺. We obtain analogous results for the space of loops of Legendrian submanifolds in (M²ⁿ⁺¹, α).