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Points fixes d’une application symplectique homologue à l’identité. (French) Zbl 0555.58013
Let (M,$$\sigma)$$ be a closed symplectic manifold, and $$(\phi_ t)$$ a Hamiltonian isotopy of M, i.e. $${\dot \phi}{}_ t=X_ t\circ \phi_ t$$ where $$X_ t$$ is a Hamiltonian vectorfield. V. I. Arnold has conjectured that $$\phi_ 1$$ has at least as many fixed points as a function on M has critical points. Using a method introduced by C. C. Conley and E. Zehnder for $$M=T^{2n}$$ [Invent. Math. 73, 33-49 (1983; Zbl 0516.58017)], we prove the following result: Assume M admits a metric of nonpositive curvature such that: 1) $$\sigma (X,Y)=<JX,Y>$$, where J is an almost complex structure on M; 2) $$(\exp^*_ p\sigma)_ X(Y,JY)\geq \alpha | Y|^ 2$$, with $$\alpha >0$$, for $$p\in M$$ and $$X,Y\in T_ pM$$. Then $$\phi_ 1$$ has at least $$CL(M)+1$$ fixed points, where $$CL(M)=cuplength$$ of $$M=\sup \{\ell: \exists$$ a ring R and $$\omega_ 1,...,\omega_{\ell}\in \tilde H^*(M;R)$$ with $$\omega_ 1\cup...\cup \omega_{\ell}\neq 0\}$$. If the fixed points are nondegenerate then their number is at least $$SB(M)=\sup \{rk H^*(M;F)$$, F a field$$\}$$. In particular this proves the Arnold conjecture for surfaces of genus $$\geq 1$$.

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 57R70 Critical points and critical submanifolds in differential topology
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