Sur les points fixes des difféomorphismes exacts symplectiques de \(T^ n\times R^ n\). (Fixed points of exact symplectic diffeomorphisms of \(T^ n\times R^ n)\). (French) Zbl 0696.57017

Arnaud, Marie-Claude

C. R. Acad. Sci., Paris, Sér. I 309, No. 3, 191-194 (1989).

Definitions. F: \(T^ n\times R^ n\mapsto T^ n\times R^ n\), (\(\theta\),r)\(\mapsto (\Theta,R)\)
1) is exact symplectic if (Rd \(\Theta\)-rd \(\theta)\) is exact;
2) is completely integrable if \(F(\theta,r)=(\theta,r+1(\theta))\) where \(1\in C^ r(T^ n,R^ n);\)
3) is weakly monotone if Det(\(\partial \Theta /\partial r)\neq 0\) on \(T^ n\times \{0\}.\)
If F: \(T^ n\times R^ n\to T^ n\times R^ n\), (\(\theta\),r)\(\mapsto (\Theta,R)\), is an exact symplectic weakly monotone \(C^ r\)- diffeomorphism, then:
1) its torsion is (\(\partial \Theta /\partial r)\); when we speak of definite torsion, torsion with signature (p,q), we will speak of definite \((\partial \Theta /\partial r+^ t\partial \Theta /\partial r)\) or signature of \((\partial \Theta /\partial r+^ t\partial \Theta /\partial r);\)
2) if F is near an exact symplectic completely integrable weakly monotone \(C^ r\)-diffeomorphism (in \(C^ 1\)-topology), then F has a radially transformed torus with equation \(\eta \in C^ r(T^ n,R^ n)\), i.e. \(\eta\) is near (in \(C^ 1\)-topology) 0 and: \(\forall \theta \in T^ n\), \(\Theta (\theta,\eta (\theta))=\theta;\)
3) then the radial function is the unique \(\phi \in C^{r+1}(T^ n,R)\) such that: \(\forall \theta \in T^ n\), \(R(\theta,\eta (\theta))-\eta (\theta)=^ td\phi (\theta)\) and \(\phi (0)=0.\)
Proposition 1. If F: \(T^ n\times R^ n\to T^ n\times R^ n\), \((\theta,r)\mapsto (\theta +F(\theta,r),R)\) is an exact symplectic \(C^ r\)-diffeomorphism near an exact symplectic completely integrable weakly monotone \(C^ r\)-diffeomorphism (in \(C^ 1\)-topology) which:
1) is generic (in \(C^ r\)-topology);
2) has its torsion (\(\partial \epsilon /\partial r)\) with signature (q,n- q) on \(T^ n\times \{0\};\)
3) is such that, if \(\eta\) is the equation of the radially transformed torus and \(\phi\) the radial function: \(\forall \theta \in T^ n\), \(\| D^ 2\phi (\theta)\| <2\| ((\partial F/\partial r)^{-1}+^ t(\partial F/\partial r)^{-1})^{-1}\|^{-1}(\theta,\eta (\theta))\); then, if \(\theta\) is a critical point of \(\phi\) such that: signature(Hessian \(\phi\) (\(\theta)\))\(=(p,n-p)\), the elliptic dimension of the fixed point (\(\theta\),\(\eta\) (\(\theta)\)) is more than \(2| p- q|.\)
Corollary. If F satisfies 1), 3) and has its torsion definite on \(T\times \{0\}\), then F has at least one completely elliptic fixed point.
Proposition 2. If \(F_ 0: T^ n\times R^ n\to T^ n\times R^ n\) is an exact symplectic weakly monotone completely integrable \(C^ r\)- diffeomorphism, then: there exists a \(C^ 1\)-open set U of exact symplectic weakly monotone \(C^ r\)-diffeomorphisms such that:
1) \(F_ 0\in Adh_ r(U);\)
2) each fixed point of each element of U has hyperbolic dimension less than 4.

Reviewer: M.-C.Arnaud

MSC:

57R50

Differential topological aspects of diffeomorphisms

Keywords:

exact symplectic; completely integrable; weakly monotone; torsion; definite torsion; signature; radial function; elliptic dimension; hyperbolic dimension