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Added the dynamical-systems and contact-geometry tags, edited spelling and formatting
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Marco Golla
Marco Golla
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non degenerate Non-degenerate periodic orbits in the boundary of Lefschetz firbationfibration over a disk

Suppose that $\pi:E \to D$ is a four dimension4-dimensional Lefschetz fibration over a disk., and let $\Omega$ isbe a closed two form2-form on $E$ such that it is non-degenerate fiberwise.

For any $x \in E$, there is a decomposition $TE_x=TE_x^h \oplus TE_x^v$, $TE^v=kerd\pi$$TE^v=\ker d\pi$ and $TE_x^h=\{v \in TE_x: \Omega(v,w)=0,\forall w \in TE_x^v\}$. Let $\partial_t$ be the coordinate vector field on $S^1=\partial D$, $R$ be the horizontal lift of $\partial_t$ and $\phi_{\Omega}$ be the flow generated by $R$. Taking base point $1 \in S^1$, then we get a map $\phi_{\Omega}: \pi^{-1}(1)\to\pi^{-1}(1)$, so the boundary $\partial E=\pi^{-1}(\partial D)$ is a mapping tourstorus $Y_{\phi_{\Omega}}$.

For the mapping tourstorus, we can define thea periodic orbit to be thean integral curve of vector field $R$. I don't know whether the periodic orbit are non degenerate.

My question: Can I do a perturbation of $\Omega$, denote it by $\Omega'$, so that the periodic orbit with period $\le d$ on the mapping tours $Y_{\phi_{\Omega'}}=\pi^{-1}(\partial D)$ are non-degenerate?

Thank you very much.

Can I do a perturbation $\Omega'$ of $\Omega$ such that the periodic orbits with period $\le d$ on the mapping torus $Y_{\phi_{\Omega'}}=\pi^{-1}(\partial D)$ are non-degenerate?

non degenerate periodic in the boundary of Lefschetz firbation over a disk

Suppose that $\pi:E \to D$ is a four dimension Lefschetz fibration over a disk. $\Omega$ is closed two form on $E$ such that it is non-degenerate fiberwise.

For any $x \in E$, there is a decomposition $TE_x=TE_x^h \oplus TE_x^v$, $TE^v=kerd\pi$ and $TE_x^h=\{v \in TE_x: \Omega(v,w)=0,\forall w \in TE_x^v\}$. Let $\partial_t$ be the coordinate vector field on $S^1=\partial D$, $R$ be the horizontal lift of $\partial_t$ and $\phi_{\Omega}$ be the flow generated by $R$. Taking base point $1 \in S^1$, then we get a map $\phi_{\Omega}: \pi^{-1}(1)\to\pi^{-1}(1)$, so the boundary $\partial E=\pi^{-1}(\partial D)$ is a mapping tours $Y_{\phi_{\Omega}}$.

For the mapping tours, we can define the periodic orbit to be the integral curve of vector field $R$. I don't know whether the periodic orbit are non degenerate.

My question: Can I do a perturbation of $\Omega$, denote it by $\Omega'$, so that the periodic orbit with period $\le d$ on the mapping tours $Y_{\phi_{\Omega'}}=\pi^{-1}(\partial D)$ are non-degenerate?

Thank you very much.

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.

For any $x \in E$, there is a decomposition $TE_x=TE_x^h \oplus TE_x^v$, $TE^v=\ker d\pi$ and $TE_x^h=\{v \in TE_x: \Omega(v,w)=0,\forall w \in TE_x^v\}$. Let $\partial_t$ be the coordinate vector field on $S^1=\partial D$, $R$ be the horizontal lift of $\partial_t$ and $\phi_{\Omega}$ be the flow generated by $R$. Taking base point $1 \in S^1$, then we get a map $\phi_{\Omega}: \pi^{-1}(1)\to\pi^{-1}(1)$, so the boundary $\partial E=\pi^{-1}(\partial D)$ is a mapping torus $Y_{\phi_{\Omega}}$.

For the mapping torus, we can define a periodic orbit to be an integral curve of vector field $R$. I don't know whether the periodic orbit are non degenerate.

Can I do a perturbation $\Omega'$ of $\Omega$ such that the periodic orbits with period $\le d$ on the mapping torus $Y_{\phi_{\Omega'}}=\pi^{-1}(\partial D)$ are non-degenerate?

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Trick1234
Trick1234
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non degenerate periodic in the boundary of Lefschetz firbation over a disk

Suppose that $\pi:E \to D$ is a four dimension Lefschetz fibration over a disk. $\Omega$ is closed two form on $E$ such that it is non-degenerate fiberwise.

For any $x \in E$, there is a decomposition $TE_x=TE_x^h \oplus TE_x^v$, $TE^v=kerd\pi$ and $TE_x^h=\{v \in TE_x: \Omega(v,w)=0,\forall w \in TE_x^v\}$. Let $\partial_t$ be the coordinate vector field on $S^1=\partial D$, $R$ be the horizontal lift of $\partial_t$ and $\phi_{\Omega}$ be the flow generated by $R$. Taking base point $1 \in S^1$, then we get a map $\phi_{\Omega}: \pi^{-1}(1)\to\pi^{-1}(1)$, so the boundary $\partial E=\pi^{-1}(\partial D)$ is a mapping tours $Y_{\phi_{\Omega}}$.

For the mapping tours, we can define the periodic orbit to be the integral curve of vector field $R$. I don't know whether the periodic orbit are non degenerate.

My question: Can I do a perturbation of $\Omega$, denote it by $\Omega'$, so that the periodic orbit with period $\le d$ on the mapping tours $Y_{\phi_{\Omega'}}=\pi^{-1}(\partial D)$ are non-degenerate?

Thank you very much.