The C⁰ distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms, and a generalisation of the C⁰ distance on the space of Riemannian metrics. I will explain the following recent result, obtained in collaboration with Lucas Dahinden, Matthias Meiwes and Abror Pirnapasov: the topological entropy of Reeb flows on contact 3-manifolds is lower semicontinuous with respect to the C⁰ metric on a C-infinity open dense set of the space of Reeb flows.
Applied to geodesic flows of Riemannian metrics on surfaces, this says that for "most" Riemannian metrics on closed surfaces, one cannot destroy positivity of topological entropy by C⁰-small perturbations of the metric. This is in some sense unexpected, as the geodesic flow depends on the derivatives of the Riemannian metric.