Positive paths in diffeomorphism groups of manifolds with a contact distribution

Abstract

Given a cooriented contact manifold $(M,\xi)$, it is possible to define a notion of positivity on the group $\mathrm{Diff}(M)$ of diffeomorphisms of $M$, by looking at paths of diffeomorphisms that are positively transverse to the contact distribution $\xi$. We show that, in contrast to the analogous notion usually considered on the group of diffeomorphisms preserving $\xi$, positivity on $\mathrm{Diff}(M)$ is completely flexible. In particular, we show that for the standard contact structure on $\mathbb{R}^{2n+1}$ any two diffeomorphisms are connected by a positive path. This result generalizes to compactly supported diffeomorphisms on a large class of contact manifolds. As an application we answer a question about Legendrians in thermodynamic phase space posed by Entov, Polterovich and Ryzhik in the context of thermodynamic processes.