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A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a Hamiltonian diffeomorphism. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying to understand Theorem 1.11 saying that the special fiber of the moment map is strongly non-displaceable. For an explicit example $S^2$ with moment map as the height function $H\colon S^2 \rightarrow [-1,1]$, the special fiber is the equator $X:= H^{-1}(0)$.

They actually prove that $X$ is superheavy, which means that $\zeta(K) \leq \sup_X K$ for all smooth function $K$ on $S^2$. Here symplectic quasi-state $\zeta$ is defined as $\zeta(K):= \lim_{r \rightarrow \infty}\frac{c(rK)}{r}$, where $c$ is the spectral invariant defined using filtered Floer homology. More explicitly, $$c(K):= \inf \{\alpha \mid PSS([S^2]) \in HF_*^{\alpha}(K)\}$$ where $PSS\colon QH_*(S^2) \rightarrow HF_*(S^2)$ is the PSS-isomorphism from quantum homology to Floer homology by Piunikhin-Salamon-Schwarz. The upper index $\alpha$ indicates that elements are coming from elements whose value of the action functional is less than $\alpha$.

According to their Proposition 4.3, if a function $K$ on $S^2$ is zero on a superheavy set, then $\zeta(K) = 0$. Because I am a beginner, I tried to compute this directly for $K=H$, but I got $\zeta(H)=1$. I must be wrong somewhere but I don't know where. The following is my computation.

I follow the sign convention of Entov and Polterovich. Take a non-integer $\epsilon>0$ so that the only orbits of period $1$ of Hamiltonian $K= \epsilon H$ are fixed points $N$ and $S$. Critical points of the action functional $$\mathcal{A}_K(\gamma, u)= \int_0^1 K(\gamma(t))dt - \int u^*\omega$$ are either $(N,u)$ or $(S,u)$, where $u\colon S^2 \rightarrow S^2$ is any map. If we compute the Conley-Zehnder index, for an integer $k$ with $k-1 < \epsilon < k$, we have $$\mu_{CZ}(N, u)= 1+k+4m, \quad \mu_{CZ}(S, u) = 1-k+4m$$ where $m$ is an integer representing the degree of $u$. So the differential is trivial and elements of $HF_*(K)$ are generated by $(N,u)$ and $(S,u)$. The image of the fundamental class by $PSS$ should be the north pole $(N, pt)$ and $\mathcal{A}_K(N,pt)= \epsilon$. We get $$c(\epsilon H) = \epsilon.$$ Taking $\epsilon \rightarrow \infty$, we have $\zeta(H) = 1$.

Where am I doing wrong?

[Added] the Conley-Zehnder index should be $$\mu_{CZ}(N, u)= 2k+4m, \quad \mu_{CZ}(S, u) = 2-2k+4m,$$ so odd degree elements doesn't appear.

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a Hamiltonian diffeomorphism. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying to understand Theorem 1.11 saying that the special fiber of the moment map is strongly non-displaceable. For an explicit example $S^2$ with moment map as the height function $H\colon S^2 \rightarrow [-1,1]$, the special fiber is the equator $X:= H^{-1}(0)$.

They actually prove that $X$ is superheavy, which means that $\zeta(K) \leq \sup_X K$ for all smooth function $K$ on $S^2$. Here symplectic quasi-state $\zeta$ is defined as $\zeta(K):= \lim_{r \rightarrow \infty}\frac{c(rK)}{r}$, where $c$ is the spectral invariant defined using filtered Floer homology. More explicitly, $$c(K):= \inf \{\alpha \mid PSS([S^2]) \in HF_*^{\alpha}(K)\}$$ where $PSS\colon QH_*(S^2) \rightarrow HF_*(S^2)$ is the PSS-isomorphism from quantum homology to Floer homology by Piunikhin-Salamon-Schwarz. The upper index $\alpha$ indicates that elements are coming from elements whose value of the action functional is less than $\alpha$.

According to their Proposition 4.3, if a function $K$ on $S^2$ is zero on a superheavy set, then $\zeta(K) = 0$. Because I am a beginner, I tried to compute this directly for $K=H$, but I got $\zeta(H)=1$. I must be wrong somewhere but I don't know where. The following is my computation.

I follow the sign convention of Entov and Polterovich. Take a non-integer $\epsilon>0$ so that the only orbits of period $1$ of Hamiltonian $K= \epsilon H$ are fixed points $N$ and $S$. Critical points of the action functional $$\mathcal{A}_K(\gamma, u)= \int_0^1 K(\gamma(t))dt - \int u^*\omega$$ are either $(N,u)$ or $(S,u)$, where $u\colon S^2 \rightarrow S^2$ is any map. If we compute the Conley-Zehnder index, for an integer $k$ with $k-1 < \epsilon < k$, we have $$\mu_{CZ}(N, u)= 1+k+4m, \quad \mu_{CZ}(S, u) = 1-k+4m$$ where $m$ is an integer representing the degree of $u$. So the differential is trivial and elements of $HF_*(K)$ are generated by $(N,u)$ and $(S,u)$. The image of the fundamental class by $PSS$ should be the north pole $(N, pt)$ and $\mathcal{A}_K(N,pt)= \epsilon$. We get $$c(\epsilon H) = \epsilon.$$ Taking $\epsilon \rightarrow \infty$, we have $\zeta(H) = 1$.

Where am I doing wrong?

[Added] the Conley-Zehnder index should be $$\mu_{CZ}(N, u)= 2k+4m, \quad \mu_{CZ}(S, u) = 2-2k+4m,$$ so odd degree elements don't appear.

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a Hamiltonian diffeomorphism. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying to understand Theorem 1.11 saying that the special fiber of the moment map is strongly non-displaceable. For an explicit example $S^2$ with moment map as the height function $H\colon S^2 \rightarrow [-1,1]$, the special fiber is the equator $X:= H^{-1}(0)$.

They actually prove that $X$ is superheavy, which means that $\zeta(K) \leq \sup_X K$ for all smooth function $K$ on $S^2$. Here symplectic quasi-state $\zeta$ is defined as $\zeta(K):= \lim_{r \rightarrow \infty}\frac{c(rK)}{r}$, where $c$ is the spectral invariant defined using filtered Floer homology. More explicitly, $$c(K):= \inf \{\alpha \mid PSS([S^2]) \in HF_*^{\alpha}(K)\}$$ where $PSS\colon QH_*(S^2) \rightarrow HF_*(S^2)$ is the PSS-isomorphism from quantum homology to Floer homology by Piunikhin-Salamon-Schwarz. The upper index $\alpha$ indicates that elements are coming from elements whose value of the action functional is less than $\alpha$.

According to their Proposition 4.3, if a function $K$ on $S^2$ is zero on a superheavy set, then $\zeta(K) = 0$. Because I am a beginner, I tried to compute this directly for $K=H$, but I got $\zeta(H)=1$. I must be wrong somewhere but I don't know where. The following is my computation.

I follow the sign convention of Entov and Polterovich. Take a non-integer $\epsilon>0$ so that the only orbits of period $1$ of Hamiltonian $K= \epsilon H$ are fixed points $N$ and $S$. Critical points of the action functional $$\mathcal{A}_K(\gamma, u)= \int_0^1 K(\gamma(t))dt - \int u^*\omega$$ are either $(N,u)$ or $(S,u)$, where $u\colon S^2 \rightarrow S^2$ is any map. If we compute the Conley-Zehnder index, for an integer $k$ with $k-1 < \epsilon < k$, we have $$\mu_{CZ}(N, u)= 1+k+4m, \quad \mu_{CZ}(S, u) = 1-k+4m$$ where $m$ is an integer representing the degree of $u$. So the differential is trivial and elements of $HF_*(K)$ are generated by $(N,u)$ and $(S,u)$. The image of the fundamental class by $PSS$ should be the north pole $(N, pt)$ and $\mathcal{A}_K(N,pt)= \epsilon$. We get $$c(\epsilon H) = \epsilon.$$ Taking $\epsilon \rightarrow \infty$, we have $\zeta(H) = 1$.

Where am I doing wrong?

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a Hamiltonian diffeomorphism. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying to understand Theorem 1.11 saying that the special fiber of the moment map is strongly non-displaceable. For an explicit example $S^2$ with moment map as the height function $H\colon S^2 \rightarrow [-1,1]$, the special fiber is the equator $X:= H^{-1}(0)$.

They actually prove that $X$ is superheavy, which means that $\zeta(K) \leq \sup_X K$ for all smooth function $K$ on $S^2$. Here symplectic quasi-state $\zeta$ is defined as $\zeta(K):= \lim_{r \rightarrow \infty}\frac{c(rK)}{r}$, where $c$ is the spectral invariant defined using filtered Floer homology. More explicitly, $$c(K):= \inf \{\alpha \mid PSS([S^2]) \in HF_*^{\alpha}(K)\}$$ where $PSS\colon QH_*(S^2) \rightarrow HF_*(S^2)$ is the PSS-isomorphism from quantum homology to Floer homology by Piunikhin-Salamon-Schwarz. The upper index $\alpha$ indicates that elements are coming from elements whose value of the action functional is less than $\alpha$.

According to their Proposition 4.3, if a function $K$ on $S^2$ is zero on a superheavy set, then $\zeta(K) = 0$. Because I am a beginner, I tried to compute this directly for $K=H$, but I got $\zeta(H)=1$. I must be wrong somewhere but I don't know where. The following is my computation.

I follow the sign convention of Entov and Polterovich. Take a non-integer $\epsilon>0$ so that the only orbits of period $1$ of Hamiltonian $K= \epsilon H$ are fixed points $N$ and $S$. Critical points of the action functional $$\mathcal{A}_K(\gamma, u)= \int_0^1 K(\gamma(t))dt - \int u^*\omega$$ are either $(N,u)$ or $(S,u)$, where $u\colon S^2 \rightarrow S^2$ is any map. If we compute the Conley-Zehnder index, for an integer $k$ with $k-1 < \epsilon < k$, we have $$\mu_{CZ}(N, u)= 1+k+4m, \quad \mu_{CZ}(S, u) = 1-k+4m$$ where $m$ is an integer representing the degree of $u$. So the differential is trivial and elements of $HF_*(K)$ are generated by $(N,u)$ and $(S,u)$. The image of the fundamental class by $PSS$ should be the north pole $(N, pt)$ and $\mathcal{A}_K(N,pt)= \epsilon$. We get $$c(\epsilon H) = \epsilon.$$ Taking $\epsilon \rightarrow \infty$, we have $\zeta(H) = 1$.

Where am I doing wrong?

[Added] the Conley-Zehnder index should be $$\mu_{CZ}(N, u)= 2k+4m, \quad \mu_{CZ}(S, u) = 2-2k+4m,$$ so odd degree elements doesn't appear.

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying to understand Theorem 1.11 saying that the special fiber of the moment map is such an example. For an explicit example $S^2$ with moment map as the height function $H\colon S^2 \rightarrow [-1,1]$, the special fiber is the equator $H^{-1}(0)$.

They actually prove that this set $X$ is superheavy, which means that $\zeta(K) \leq \sup_X K$ for all smooth function $K$ on $S^2$. Here symplectic quasi-state $\zeta$ is defined as $\zeta(K):= \lim_{r \rightarrow \infty}\frac{c(rK)}{r}$, where $c$ is the spectral invariant defined using filtered Floer homology. More explicitly, $$c(K):= \inf \{\alpha \mid PSS([S^2]) \in HF_*^{\alpha}(K)\}$$ where $PSS\colon QH_*(S^2) \rightarrow HF_*(S^2)$ is the PSS-isomorphism from quantum homology to Floer homology by Piunikhin-Salamon-Schwarz. The upper index $\alpha$ indicates that elements are coming from elements whose value of the action functional is less than $\alpha$.

According to their Proposition 4.3, if a function $K$ on $S^2$ is zero on a superheavy set, then $\zeta(K) = 0$. Because I am a beginner, I tried to compute this directly for $K=H$, but I got $\zeta(H)=1$. I must be wrong somewhere but I don't know where. The following is my computation.

I follow the sign convention of Entov and Polterovich. Take a non-integer $\epsilon>0$ so that the only orbits of period $1$ of Hamiltonian $K= \epsilon H$ are fixed points $N$ and $S$. Critical points of the action functional $$\mathcal{A}_K(\gamma, u)= \int_0^1 K(\gamma(t))dt - \int u^*\omega$$ are either $(N,u)$ or $(S,u)$, where $u\colon S^2 \rightarrow S^2$ is any map. If we compute the Conley-Zehnder index, for an integer $k$ with $k-1 < \epsilon < k$, we have $$\mu_{CZ}(N, u)= 1+k+4m, \quad \mu_{CZ}(S, u) = 1-k+4m$$ where $m$ is an integer representing the degree of $u$. So the differential is trivial and elements of $HF_*(K)$ are generated by $(N,u)$ and $(S,u)$. The image of the fundamental class by $PSS$ should be the north pole $(N, pt)$ and $\mathcal{A}_K(N,pt)= \epsilon$. We get $$c(\epsilon H) = \epsilon.$$ Taking $\epsilon \rightarrow \infty$, we have $\zeta(H) = 1$.

Where am I doing wrong?

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a Hamiltonian diffeomorphism. I am reading "Rigid subsets of symplectic manifolds" by Entov and Polterovich and trying to understand Theorem 1.11 saying that the special fiber of the moment map is strongly non-displaceable. For an explicit example $S^2$ with moment map as the height function $H\colon S^2 \rightarrow [-1,1]$, the special fiber is the equator $X:= H^{-1}(0)$.

They actually prove that $X$ is superheavy, which means that $\zeta(K) \leq \sup_X K$ for all smooth function $K$ on $S^2$. Here symplectic quasi-state $\zeta$ is defined as $\zeta(K):= \lim_{r \rightarrow \infty}\frac{c(rK)}{r}$, where $c$ is the spectral invariant defined using filtered Floer homology. More explicitly, $$c(K):= \inf \{\alpha \mid PSS([S^2]) \in HF_*^{\alpha}(K)\}$$ where $PSS\colon QH_*(S^2) \rightarrow HF_*(S^2)$ is the PSS-isomorphism from quantum homology to Floer homology by Piunikhin-Salamon-Schwarz. The upper index $\alpha$ indicates that elements are coming from elements whose value of the action functional is less than $\alpha$.

According to their Proposition 4.3, if a function $K$ on $S^2$ is zero on a superheavy set, then $\zeta(K) = 0$. Because I am a beginner, I tried to compute this directly for $K=H$, but I got $\zeta(H)=1$. I must be wrong somewhere but I don't know where. The following is my computation.

I follow the sign convention of Entov and Polterovich. Take a non-integer $\epsilon>0$ so that the only orbits of period $1$ of Hamiltonian $K= \epsilon H$ are fixed points $N$ and $S$. Critical points of the action functional $$\mathcal{A}_K(\gamma, u)= \int_0^1 K(\gamma(t))dt - \int u^*\omega$$ are either $(N,u)$ or $(S,u)$, where $u\colon S^2 \rightarrow S^2$ is any map. If we compute the Conley-Zehnder index, for an integer $k$ with $k-1 < \epsilon < k$, we have $$\mu_{CZ}(N, u)= 1+k+4m, \quad \mu_{CZ}(S, u) = 1-k+4m$$ where $m$ is an integer representing the degree of $u$. So the differential is trivial and elements of $HF_*(K)$ are generated by $(N,u)$ and $(S,u)$. The image of the fundamental class by $PSS$ should be the north pole $(N, pt)$ and $\mathcal{A}_K(N,pt)= \epsilon$. We get $$c(\epsilon H) = \epsilon.$$ Taking $\epsilon \rightarrow \infty$, we have $\zeta(H) = 1$.

Where am I doing wrong?