On the degenerated Arnold-Givental conjecture

Abstract

We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, \omega, \tau)$ with nonempty and compact real part $L={\rm Fix}(\tau)$. For given $\Lambda\in (0, +\infty]$ and $m\in\N\cup\{0\}$ we show the equivalence of the following two claims: (i) $\sharp(L\cap\phi^H_1(L))\ge m$ for any Hamiltonian function $H\in C_0^\infty([0, 1]\times M)$ with Hofer's norm $\|H\|<\Lambda$; (ii) $\sharp {\cal P}(H,\tau)\ge m$ for every $H\in C^\infty_0(\R/\Z\times M)$ satisfying $H(t,x)=H(-t,\tau(x))\;\forall (t,x)\in\mathbb{R}\times M$ and with Hofer's norm $\|H\|<2\Lambda$, where ${\cal P}(H, \tau)$ is the set of all $1$-periodic solutions of $\dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=\tau(x(t))\;\forall t\in\R$ (which are also called brake orbits sometimes). Suppose that $(M, \omega)$ is geometrical bounded for some $J\in{\cal J}(M,\omega)$ with $\tau^\ast J=-J$ and has a rationality index $r_\omega>0$ or $r_\omega=+\infty$. Using Hofer's method we prove that if the Hamiltonian $H$ in (ii) above has Hofer's norm $\|H\|<r_\omega$ then $\sharp(L\cap\phi^H_1(L))\ge\sharp {\cal P}_0(H,\tau)\ge {\rm Cuplength}_{\F}(L)$ for $\F=\Z_2$, and further for $\F=\Z$ if $L$ is orientable, where ${\cal P}_0(H,\tau)$ consists of all contractible solutions in ${\cal P}(H,\tau)$.