Àííîòàöèÿ:
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.
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