My primary interests and areas of research are

- Topological Hamiltonian and contact dynamics
- Symplectic and contact geometry and topology
- Topological dynamics

and the deeper topological structure underlying symplectic and contact topology. *Topological Hamiltonian dynamics* and *topological contact dynamics* are *natural* and *genuine extensions* of smooth Hamiltonian and contact dynamics to topological dynamics, and of symplectic and contact diffeomorphisms to homeomorphisms as transformations preserving the additional geometric structure. These new theories have numerous applications to their smooth counterparts, as well as to other areas of mathematics, such as *topological dynamics*, in particular in *low dimensions*, and to *geodesic flows* on Riemannian manifolds. Applications are also expected to *billiard dynamics*. I also have an interest in *action selectors* in *Hamiltonian Floer theory* and related subjects.

A *topological Hamiltonian* or *contact dynamical system* consists of a *topological Hamiltonian* or *contact isotopy* (a *continuous* isotopy of *homeomorphisms*), together with a possibly *non-smooth topological Hamiltonian function* on the underlying symplectic or contact manifold, and a *topological conformal factor* (a *continuous* function) in the contact case. By definition, this Hamiltonian function is the *limit* of a sequence of smooth Hamiltonian functions with respect to the usual *Hofer metric*. Moreover, the corresponding sequence of smooth Hamiltonian or contact isotopies *converges uniformly* to the above continuous isotopy, and their smooth conformal factors *converge uniformly* to the above continuous function associated to the limit isotopy. In other words, we consider limits of smooth Hamiltonian and contact dynamical systems with respect to a metric that combines *topological* and *dynamical information*. Similarly, *topological automorphisms* of a *symplectic* or *contact structure* or *form* are C^0-limits of symplectic or contact diffeomorphisms (together with their conformal factors in the contact case).

A topological Hamiltonian or contact isotopy is *not* generated by a vector field, and may *not* even be Lipschitz *continuous*; nonetheless, it is *uniquely determined* by its associated topological Hamiltonian function, and in turn, in the contact case it *determines uniquely* its topological conformal factor. *Composition* and *inversion* can therefore be defined as in the smooth case, and the usual *transformation law* continues to hold. Conversely, every topological Hamiltonian or contact isotopy *possesses* a *unique* topological Hamiltonian function. The topological automorphism groups of a contact structure and a contact form exhibit surprising *rigidity* properties analogous to the well-known Eliashberg-Gromov rigidity in the case of a symplectic structure.

The uniqueness theorems for topological Hamiltonian and contact dynamical systems, rigidity, the group structures, and the transformation law, provide ample evidence that Hamiltonian and contact dynamics are in fact *topological theories*, and a priori smooth invariants can be extended to topological Hamiltonian and contact dynamical systems. The study of these topological theories is to a large extend motivated by *physical considerations* and by various *continuity phenomena* already present in their smooth counterparts, and in turn, they have *applications* to *smooth* Hamiltonian and contact dynamics as well as to *topological dynamics* and *low-dimensional topology*. For example, one can answer important questions of V. I. Arnold regarding the *continuity* of the *helicity* of a *volume preserving isotopy*, and its behavior under *conjugation* by *volume preserving homeomorphisms*, provided the isotopy and the transformation can be described as *lifts* from a manifold one dimension lower. The relation between topological Hamiltonian dynamics and topological contact dynamics is … [More]