Combinatorial structures of the space of Hamiltonian vector fields on compact surfaces

Abstract

In the time evolution of vector fields, the topologies of vector fields can be changed by creations and annihilations of singular points and by switching combinatorial structures of separatrices. In this paper, to describe the possible generic time evolution of Hamiltonian vector fields on surfaces with or without restriction conditions, we study the hierarchical structure of the space of such vector fields under the non-existence of creations and annihilations of singular points and the non-existence of fake multi-saddles. More precisely, consider Hamiltonian vector fields with finitely many singular points but without fake multi-saddles on a compact surface. Then the space of topologically equivalent classes of such vector fields is a finite abstract cell complex such that the codimension of a cell corresponds to the instability of a Hamiltonian vector field. In other words, any k-cell corresponds to a "codimension k" topologically equivalent class of Hamiltonian vector fields.

Combinatorial structures of the space of gradient vector fields on compact surfaces

“…Notice that the intermediate flow with a limit circuit must appear between Morse flows and non-Morse Morse-Smale-like flows in l≥0 Q r k−/2,k+/2,l,>−1 (S). To state such transitions via saddle connections containing non-periodic limit circuits, we need the "codimension" of the multisaddle connection diagram for flows in the space l≥0 Q r k−/2,k+/2,l, * * (S) like the "codimension" of the multi-saddle connection diagram for Hamiltonian flows [20]. We will report such transitions in the near future.…”

“…On the other hand, the topologies of such fluids also can be changed by switching combinatorial structures of separatrices. Such combinatorial structures are studied from fluid mechanics [2,7,11], integrable systems [5], and dynamical systems [8,[14][15][16][17][20][21][22][23].…”