Language：English   Publishing type：Research paper (scientific journal)  

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.

arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we obtain a family of singular Lagrangian submanifolds in [Formula: see text] with various rigidities.

DOI： 10.1142/s0219199720500479

arXiv

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Language：English   Publishing type：Research paper (scientific journal)  

We show that the presence of a non-contractible one-periodic orbit of a
Hamiltonian diffeomorphism of a connected closed symplectic manifold
$(M,\omega)$ implies the existence of infinitely many non-contractible simple
periodic orbits, provided that the symplectic form $\omega$ is aspherical and
the fundamental group $\pi_1(M)$ is either a virtually abelian group or an
$\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian
diffeomorphisms of closed monotone or negative monotone symplectic manifolds
under the same conditions on their fundamental groups. These results generalize
some works by Ginzburg and G\"urel. The proof uses the filtered Floer--Novikov
homology for non-contractible periodic orbits.

DOI： 10.4310/JSG.2019.v17.n6.a9

arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding [Formula: see text] with respect to the fragmentation norm on the group [Formula: see text] of Hamiltonian diffeomorphisms of a symplectic manifold [Formula: see text]. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on [Formula: see text], where [Formula: see text] is a closed Riemannian surface of genus [Formula: see text].

DOI： 10.1142/S179352532050017X

arXiv

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：Japanese

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In this paper, we show that the pointwise finite-dimensional two-parameter
persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of
interlevel filtered Floer homology, is rectangle-decomposable. This allows for
the definition of a barcode $\mathcal{B}_*^{(\bullet,\bullet]}$ consisting
solely of rectangles in $\mathbb{R}^2$ associated with
$\mathbb{HF}_*^{(\bullet,\bullet]}$. We observe that this rectangle barcode
contains information about Usher's boundary depth and spectral invariants
developed by Oh, Schwarz, and Viterbo. Moreover, we introduce a novel invariant
extracted from $\mathcal{B}_*^{(\bullet,\bullet]}$, which proves instrumental
in detecting periodic solutions of Hamiltonian systems. Additionally, we
establish relevant stability results, particularly concerning the bottleneck
distance and Hofer's distance.

arXiv

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Other Link： http://arxiv.org/pdf/2312.07847v2

We study Farber's topological complexity for monotone symplectic manifolds.
More precisely, we estimate the topological complexity of 4-dimensional
spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

arXiv

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Other Link： http://arxiv.org/pdf/2311.14989v1