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On the action spectrum for closed symplectically aspherical manifolds. (English) Zbl 1023.57020
Let $$(M, \omega)$$ be a symplectic manifold where the symplectic form $$\omega$$ and the Chern class $$c_1(M)$$ vanish on $$\pi_2(M)$$, that is, $$(M, \omega)$$ is symplectically aspherical. The present paper considers a version of Floer homology on $$(M, \omega)$$. Let $$\Omega(M)^0\subset C^{\infty} (\mathbb R/\mathbb Z, M)$$ be the space of contractible free loops. To any given Hamiltonian $$H: [0, 1]\times M\to\mathbb R$$, the action functional $$\mathcal A_H= \int_{D^2}\overline x^*\omega-\int_{S^1}H(t, x) dt$$ is associated, where $$\overline x: D^2\to M$$ is any extension of $$x\in \Omega^0(M)$$ to the unit disk. By P. Seidel [Geom. Funct. Anal. 7, 1046-1095 (1997; Zbl 0928.53042)], the author shows that the set of critical values of $$\mathcal A_H$$, called the action spectrum, depends only on the associated Hamiltonian diffeomorphism $$\phi=\phi^1_H$$. A well-defined action spectrum $$\Sigma_{\phi}$$ provides the action spectrum bundle $$\Sigma\to \text{Ham}(M, \omega)$$.
The main result (Theorem 1.2) says that for each nonzero cohomology class $$\alpha\in H^*(M)$$, there exists a section $$c(\alpha)$$ of the action spectrum bundle which is continuous with respect to the Hofer-metric on $$\text{Ham}(M, \omega)$$ and satisfies the following properties.
1) $$c(\lambda\alpha)=c(\alpha)$$ for all $$\lambda\in \mathbb R$$ and $$\alpha\in H^*(M)$$ with $$\lambda\neq 0$$.
2) $$c([M])\leq c(\alpha)\leq c(1)$$ for all $$0\neq\alpha\in H^*(M)$$ where $$[M]=[\omega^n]\in H^{2n}(M)$$ and $$1\in H^0(M)$$ and if $$\alpha\in H^k(M)$$ for $$0<k<2n$$ then we have strict inequality over the regular automorphisms.
3) $$c(1; \phi)-c([M]; \phi)=\gamma(\phi)\leq d_H(\phi, id)=\inf\{\|H\|\mid\phi= \phi^1_H\}$$, $$\phi\in \text{Ham}(M, \omega)$$, where $$\|H\|=\int^1_o \sup H(t, \cdot) dt- \int^1_o \inf H(t, \cdot) dt$$.
4) If $$\alpha\cup\beta\neq 0$$, then $$c(\alpha\cup\beta; \psi\circ\phi) \leq c(\alpha; \phi)+c(\beta; \psi)$$ for all $$\phi, \psi\in \text{Ham}(M, \omega)$$.
5) $$c([M]; \phi)=-c(1; \phi^{-1})$$ for all $$\phi\in \text{Ham}(M,\omega)$$.
As a result, $$d(\phi, \psi)=\gamma(\phi\psi^{-1})$$ is a bi-invariant metric bounded above by Hofer metric. The author claims that $$\gamma$$ measures the maximal action difference of “homologically visible” $$1$$-periodic solutions of the Hamiltonian equations. Based on $$\gamma$$, one defines a symplectic capacity for subsets of $$(M, \omega)$$ which is bounded below by the Hofer-Zehnder capacity and less than or equal to twice the displacement energy of F. Lalonde and D. McDuff [Ann. Math. 141, 349-371 (1995; Zbl 0829.53025)]. Then it is shown that a Hamiltonian automorphism whose support has finite capacity has infinitely many nontrivial geometrically distinct periodic points corresponding to contractible periodic solutions. The method of the author is closely related to the question of the diameter of $$\text{Ham}(M, \omega)$$ in the Hofer-metric.

##### MSC:
 57R58 Floer homology 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 53D40 Symplectic aspects of Floer homology and cohomology
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