∞-jets of diffeomorphisms preserving orbits of vector fields

Abstract

Let F be a C ∞ vector field defined near the origin O ∈ R , F (O) = 0, and (F ) be its local flow. Denote bŷ (F ) the set of germs of orbit preserving diffeomorphisms : R → R at O, and letˆ id (F ) , ( ≥ 0), be the identity component ofˆ (F ) with respect to the weak Whitney W topology. Thenˆ id (F ) ∞ contains a subset S (F ) consisting of maps of the form F α( ) ( ), where α : R → R runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, thenŜ (F ) =ˆ id (F ) 0In this paper we present a class of examples of vector fields with degenerate singularities at O for whichŜ (F ) formally coincides withˆ id (F ) 1 , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and "reduced" Hamiltonian vector fields of real homogeneous polynomials in two variables. MSC:37C10

Connected components of partition preserving diffeomorphisms

“…This will imply that LS(f ) is isomorphic with a finite subgroup of SO(2), and therefore is cyclic. Also notice that the fact that LS(f ) is discrete also follows from [12]. First we establish the following three statements: Claim 6.3.…”

“…Thus S id (f ) r consists of diffeomorphisms h ∈ S(f ) isotopic in S(f ) to id R 2 via (an f -preserving isotopy) H : R 2 × I → R 2 whose partial derivatives in (x, y) ∈ R 2 up to order r continuously depend on (x, y, t), see Section 2 for a precise definition. Then it is easy to see that It follows from results [12,13] that S id (f ) ∞ = S id (f ) 1 . Moreover, it is actually proved in [9] that S id (f ) ∞ = S id (f ) 0 for p ≤ 2, see also [11].…”

Symmetries of center singularities of plane vector fields

“…We have S h.F / D E C .F / and that local sections of preserve smoothness. These two assumptions imply parameter rigidity of F by results of [30].…”

“…Recall that a vector field F is parameter rigid if, for any other vector field F 0 such that every orbit o 0 of F 0 is included in some orbit of F; there exists a C 1 -function W D 2 ! R such that F 0 D F; see [29,30].…”

Deformations of circle-valued Morse functions on surfaces

“…The following theorem is proved in [34] for Morse maps and in [40] for all f ∈ F(M, P ). In fact, it is a consequence of a series of papers [32,33,38,41,42] about diffeomorphisms preserving orbits of flows. The technique developed in those paper is also extensively used for the proof of almost all presented here results, e.g.…”