Topological contact dynamics II: Topological automorphisms, contact homeomorphisms, and non-smooth contact dynamical systems

This sequel to our previous paper continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. This article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is arbitrarily close to a continuous contact or strictly contact dynamical system with the same end point. In particular, the above groups of time-one maps are independent of the choice of norm in the definition of the contact distance. On every contact manifold we construct topological contact dynamical systems with time-one maps that fail to be Lipschitz continuous, and smooth contact vector fields whose flows are topologically conjugate but not conjugated by a contact $C^1$-diffeomorphism.