Transitions of Hamiltonian flows in spherical domains with finitely many punctured and complete invariants of flows of finite type

This talk is based on a following question: What is a generic transition of flows on compact surfaces? One of goal of this talk is describing an answer for Hamiltonian case. Indeed, it’s known that Hamiltonian flows on a compact surface $M$ which are structurally stable in the set $\mathcal H$ of Hamiltonian flows on $M$ form an open dense subset of $\mathcal H$. Hence we need “suitable” unstable Hamiltonian flows between structurally stable Hamiltonian flows to describe a time evolution of time dependent Hamiltonian flows (e.g. a solution of Navier-Stokes equation). In other words, we need a subset of $\mathcal H$ in which reasonable transitions are generic to describe “suitable” generic transitions. Thus we introduce a classification of evaluations and “natural” transitions to describe time evaluations of Hamiltonian flows. In particular, we give some examples to understand what are transitions. Moreover, we introduce a complete invariant which is a pair of a word and a combinatorial structure, called a COT representation and a linking structure, for more general flows (e.g. slices of flows on three dimensional manifolds) to construct a foundation of transitions of general flows on surfaces. In particular, we illustrate the invariant using Hamiltonian flows and Morse-Smale flows. If time allows, we explain open problems.