Dynamically exotic contact spheres in dimensions $\geq 7$

Alves, Marcelo R. R.; Matthias, Meiwes,

doi:10.4171/cmh/468

Cited by 15 publications

(24 citation statements)

References 33 publications

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“…Remark 2 The construction of the desired pseudo-metric in Theorem A is influenced by the concept, relative growth rate, that was first studied in details in the contact geometry set-ups in [10]. For its precise definition, see Sect.…”

Section: Resultsmentioning

confidence: 99%

“…However, our definition of d CBM in Definition 4 involves the action of group Cont 0 (M, ξ), which results in a non-trivial comparison between functions in C ∞ (M, R). For the precise formula, see (28) in (2) in Remark 11. Then the large geometric property of the pseudo-metric space (O (M,ξ ) (α 0 ), d CBM ) becomes much less obvious.…”

Section: Resultsmentioning

confidence: 99%

“…In this case, there is a geometric perspective to view any element α ∈ O (M,ξ ) (α 0 ) (cf. Section 3.2 in [2]). The completion Ŵ decomposes into SM Core( Ŵ ).…”

Section: Denote Bymentioning

confidence: 99%

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Relative growth rate and contact Banach–Mazur distance

Rosen

Zhang

2021

Geom Dedicata

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In this paper, we define non-linear versions of Banach-Mazur distance in the contact geometry set-up, called contact Banach-Mazur distances and denoted by d CBM . Explicitly, we consider the following two set-ups, either on a contact manifold W × S 1 where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of d CBM are different in these two cases. In the former case the inputs are (contact) star-shaped domains of W × S 1 which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach-Mazur distance d CBM defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg-Polterovich. The main results are the large-scale geometric properties in terms of d CBM . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou-Kashiwara-Schapira's sheaf quantization.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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Relative growth rate and contact Banach–Mazur distance

Rosen

Zhang

2021

Geom Dedicata

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“…See also [16] and [18] for related topics on the slow volume growth. Recently, Alves and Meiwes studied a volume growth of Reeb flows on contact manifolds in [3].…”

Section: Introductionmentioning

confidence: 99%

Volume growth in the component of fibered twists

Kim

Kwon

Lee³

2018

Commun. Contemp. Math.

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For a Liouville domain W whose boundary admits a periodic Reeb flow, we can consider the connected component [τ ] ∈ π0(Symp c ( W )) of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, of the component [τ ] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [τ ] has infinite order in π0(Symp c ( W )) if there is an admissible Lagrangian L in W whose wrapped Floer homology is infinite dimensional.We apply our results to fibered twists coming from the Milnor fibers of A k -type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse-Bott spectral sequences.

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“…A large class of contactomorphisms are those that arise via Reeb flows and there is an abundance of contact manifolds for which the topological entropy or the exponential orbit growth rate is positive for all Reeb flows. Examples and dynamical properties of those manifolds are investigated in [1,2,3,4,5,8,28,37]. Some of these results generalise to positive contactomorphisms [20,19], and results on the dependence of some lower bounds on topological entropy with respect to their positive contact Hamiltonians have been obtained in [21].…”

mentioning

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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Dynamically exotic contact spheres in dimensions $\geq 7$

Cited by 15 publications

References 33 publications

Relative growth rate and contact Banach–Mazur distance

Relative growth rate and contact Banach–Mazur distance

Volume growth in the component of fibered twists

Braid stability and the Hofer metric

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