Bounds on spectral norms and barcodes

Kislev, Asaf; Shelukhin, Egor

doi:10.48550/arxiv.1810.09865

Cited by 13 publications

(48 citation statements)

References 97 publications

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“…The same is true of the spectral norm γ and its variant γ ext on L L 0 (M ). This follows from the proof of Theorem E in [KS18]. When M = T * L 0 , and γ is extended to L e (T * L 0 ) ∩ L m(1,0) (T * L 0 ), the same stay true.…”

Section: Iia J-adapted Metricsmentioning

confidence: 58%

“…(d F = γ ext ): This is a variant of the usual spectral norm, as defined in [KS18], where it is also shown that it is a metric. We also have that L (M ) ⊆ L L 0 (M ) and F = F = ∅, but we only ask that L 0 ∈ L we (M ).…”

Section: J-adapted Metrics On Collections Of Lagrangian Submanifoldsmentioning

confidence: 99%

“…We remark that the polygons which are found in the proof of Theorem 5.0.2 of [BCS18] and Theorem E in [KS18] are J-holomorphic for a very specific J. This choice is made to allow for the use Lelong's inequality.…”

Section: Iia J-adapted Metricsmentioning

confidence: 99%

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Convergence and Riemannian bounds on Lagrangian submanifolds

Jean-Philippe¹

2021

Preprint

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We consider collections of Lagrangian submanifolds of a given symplectic manifold which respect uniform bounds of curvature type coming from an auxiliary Riemannian metric. We prove that, for a large class of metrics on these collections, convergence to an embedded Lagrangian submanifold implies convergence to it in the Hausdorff metric. This class of metrics includes well-known metrics such as the Lagrangian Hofer metric, the spectral norm and the shadow metrics introduced by Biran, Cornea and Shelukhin [BCS18]. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball.

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Section: Iia J-adapted Metricsmentioning

confidence: 58%

Section: J-adapted Metrics On Collections Of Lagrangian Submanifoldsmentioning

confidence: 99%

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Convergence and Riemannian bounds on Lagrangian submanifolds

Jean-Philippe¹

2021

Preprint

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“…The persistence method has been widely applied to symplectic and contact geometry. After the pioneering work by Polterovich-Shelukhin [PS16], persistence modules have been used for the study of barcodes of Floer cohomology complexes in Usher-Zhang [UZ16] and the study of spectral norms in Kislev-Shelukhin [KS18b], to name a few. In this paper, we also investigate the relation between the interleaving-like distance d D(M ) and the spectral norm (see Theorem 5.8).…”

Section: Related Workmentioning

confidence: 99%

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Asano¹,

Ike²

2022

Preprint

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We investigate sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving-like distance and construct a sheaf quantization of a hameomorphism. We also develop Lusternik-Schnirelmann theory in the microlocal theory of sheaves. With these new tools, we prove an Arnold-type theorem for the image of the zero-section under a hameomorphism by a purely sheaf-theoretic method.

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“…One direction of investigation to better understand d Hofer is to investigate how dynamical properties of Hamiltonian diffeomorphisms vary under perturbations with respect to d Hofer . This has been pursued by several authors, see [16,36,43,44,46]. Floer homology together with its action filtration behaves in a very stable way under perturbations with respect to d Hofer .…”

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confidence: 99%

Braid stability and the Hofer metric

Alves¹,

Matthias²

2021

Preprint

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In this article we show that the braid type of a set of 1periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric d Hofer . We call this new phenomenon braid stability for the Hofer metric.We apply braid stability to study the stability of the topological entropy h top of Hamiltonian diffeomorphisms on surfaces with respect to small perturbations with respect to d Hofer . We show that h top is lower semicontinuous on the space of Hamiltonian diffeomorphisms of a closed surface endowed with the Hofer metric, and on the space of compactly supported diffeormophisms of the two-dimensional disk D endowed with the Hofer metric. This answers the two-dimensional case of a question of Polterovich.En route to proving the lower semicontinuity of h top with respect to d Hofer , we prove that the topological entropy of a diffeomorphism φ on a compact surface can be recovered from the topological entropy of the braid types realised by the periodic orbits of φ.

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Bounds on spectral norms and barcodes

Cited by 13 publications

References 97 publications

Convergence and Riemannian bounds on Lagrangian submanifolds

Convergence and Riemannian bounds on Lagrangian submanifolds

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Braid stability and the Hofer metric

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