Positive topological entropy of positive contactomorphisms

Dahinden, Lucas

doi:10.4310/jsg.2020.v18.n3.a3

Cited by 4 publications

(5 citation statements)

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“…It is natural to ask if one can generalize the results of this paper to the class of positive contactomorphisms. For this, one would need to combine the ideas introduced here with the techniques developed by Dahinden, who used Rabinowitz–Floer homology to study the topological entropy of positive contactomorphisms in [18, 19].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Reeb orbits that force topological entropy

Alves¹,

Pirnapasov²

2021

Ergod. Th. Dynam. Sys.

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We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Reeb orbits that force topological entropy

Alves¹,

Pirnapasov²

2021

Ergod. Th. Dynam. Sys.

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“…We will also need to understand maps Ψ G and Ψ G geometrically. Since these maps are induced by the continuation maps Ψ G,J s t and Ψ G, J s t this can be obtained via the definition of continuation maps in (20). Indeed, it follows from the definitions of these maps, that if…”

Section: We Now Explain How To Show Thatmentioning

confidence: 99%

“…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]. A related discussion and results on questions of C 0 -stability of the topological entropy of geodesic flows can be found in [6].…”

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confidence: 98%

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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“…As a group, v-shaped wrapped Floer homology defined here is equivalent to the Lagrangian Floer homology for a trivial cobordism defined in [9] (up to minor conventional differences). Moreover, in [10,Section 3.3.2], it is shown that the v-shaped wrapped Floer homology is isomorphic to Lagrangian Rabinowitz Floer homology. This is parallel to the fact that v-shaped symplectic homology is isomorphic to Rabinowitz Floer homology as groups.…”

Section: ȟWmentioning

confidence: 99%

“…This can be regarded as a relative version of v-shaped symplectic homology introduced in [8]. We remark that v-shaped wrapped Floer homology is group isomorphic to wrapped Floer homology of trivial Lagrangian cobordisms [9] and to Lagrangian Rabinowitz Floer homology [10,22].…”

Section: Introductionmentioning

confidence: 97%

A computation of the ring structure in wrapped Floer homology

Bae

Kwon

2019

Preprint

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We give an explicit computation of the ring structure in wrapped Floer homology of real Lagrangians in A k -type Milnor fibers. In the A k -type plumbing description, those Lagrangians correspond to cotangent fibers or diagonal Lagrangians. The main ingredient of the computation is an idea of the Seidel representation. For a technical reason, we first carry out computations in v-shaped wrapped Floer homology, and this in turn gives the desired ring structure via the Viterbo transfer map.

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Positive topological entropy of positive contactomorphisms

Cited by 4 publications

References 0 publications

Reeb orbits that force topological entropy

Reeb orbits that force topological entropy

Braid stability and the Hofer metric

A computation of the ring structure in wrapped Floer homology

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