Abstract
We prove that the spaces of $C^1$ symplectomorphisms and of $C^1$ volume-preserving diffeomorphisms of connected manifolds contain residual subsets of diffeomorphisms whose centralizers are trivial.
Highlights
Let M be a connected compact manifold
Vago [BCVW] we answered negatively to Questions 1 c) in the dissipative and in the symplectic case: for any compact symplectic manifold M, there exists a familly of symplectomorphisms of M with large centraliser Z1(f ) ∩ Symp1(M ) and that is dense in a non-empty open subset U of Symp1(M ); with a different method we constructed a family of diffeomorphisms of M with large centraliser Z1(f ) that is dense in a non-empty open subset U of Diff1(M )
Kopell’s proof in [Ko] that the set of diffeomorphisms having a trivial centralizer is open and dense in Diffr(S1) for r ≥ 2 uses the fact that a C2 diffeomorphism f of [0, 1] without fixed points in (0, 1) has bounded distortion, meaning: for any x, y ∈ (0, 1), the ratio
Summary
Let M be a connected compact manifold. The centralizer of a Cr diffeomorphism f ∈ Diffr(M ) is defined as. In a recent work [BCW2] we solved Questions 1 a) and b) in the C1 dissipative case: the set of diffeomorphisms f whose centralizer Z1(f ) is trivial is a residual subset of Diff1(M ). Vago [BCVW] we answered negatively to Questions 1 c) in the dissipative and in the symplectic case: for any compact symplectic manifold M , there exists a familly of symplectomorphisms of M with large centraliser Z1(f ) ∩ Symp1(M ) and that is dense in a non-empty open subset U of Symp1(M ); with a different method we constructed a family of diffeomorphisms of M with large centraliser Z1(f ) that is dense in a non-empty open subset U of Diff1(M ) The existence of such examples in the volume-preserving setting in dimension larger or equal to 3 remains open. The conservative cases (b) and (c) of Theorem B can be derived as for case (a) with an extension result presented in section 2: any perturbation of the dynamics of a symplectomorphism inside the stable manifold of a periodic point can be realized as a perturbation of the dynamics on M
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