## 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

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