Dynamical systems on the Liouville plane and the related strictly contact systems

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. We discuss, in addition, their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given.