Entropy collapse versus entropy rigidity for Reeb and Finsler flows

Abbondandolo, Alberto; Alves, Marcelo R. R.; Sağlam, Murat; Schlenk, Félix

doi:10.48550/arxiv.2103.01144

Cited by 3 publications

(10 citation statements)

References 66 publications

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“…Structural stability asserts that a diffeomorphism T ′ that is sufficiently C 1 -close to T contains the same symbolic dynamics as that of T , and has at least the topological entropy as T . 1 Horseshoes are prevalent in dynamical systems of complex orbit structure. This is in particular the case for surface diffeomorphisms: By a celebrated result of Katok [37], any diffeomorphism of positive topological entropy has a hyperbolic fixed point with a transverse homoclinic point and has a horseshoe in some iterate.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Hofer's geometry and topological entropy

Chor¹,

Matthias²

2021

Preprint

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In this article we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps introduced by Polterovich and Shelukhin. A crucial ingredient comes from some enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Those bounds are obtained within the framework of Le Calvez and Tal's forcing theory. 2020 Mathematics Subject Classification. 37E30, 37J46. A. Chor was partially supported by the Israel Science Foundation grant 667/18. M. Meiwes was partially supported by the Israel Science Foundation grant 2026/17. 1 In fact a similar result holds for C 0 -perturbations of T , see [45].1 * (φ) α for r ∈ Ê, and whose linear maps π r,s : HF r * (φ) α → HF s * (φ) α are induced by the inclusion maps CF r * (M, H) α → CF s * (M, H) α , where H is a Hamiltonian that generates φ.

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Section: Introduction and Resultsmentioning

confidence: 99%

Hofer's geometry and topological entropy

Chor¹,

Matthias²

2021

Preprint

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“…) for the values of a ∈ [0, +∞) on which the moduli spaces of Floer cylinders for (Q a , J s t (a)) are not regularly cut out. The main thing to be observed is that for every a ∈ [0, +∞) the Floer cylinders of (Q a , J s t (a)) satisfy an estimate similar 1 to the one in (31). More precisely if u Floer cylinder used in the definition of S and γ is the negative asymptotic limit of u and γ ′ is its positive asymptotic limit, than we have ( 45)…”

Section: We Now Explain How To Show Thatmentioning

confidence: 90%

“…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].…”

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“…Recall that a Hamiltonian H : S 1 ×Σ → Ê is called normalized if Σ H t ω = 0 for each t ∈ S 1 , where H t (•) := H(t, •). Given φ ∈ Ham(Σ, ω) it is always possible to find a normalized Hamiltonian H which generates φ.…”

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confidence: 99%

“…If Σ = S 2 , we fix for each free homotopy class of loops α ∈ [S 1 , M] a representative η α : S 1 → Σ of α. If y is a smooth non-contractible loop and α its free homotopy class, we say that a smooth mapping w y : [0, 1] × S 1 → Σ with w y (0, t) = η α (t) and w y (1, t) = y(t) is a cylindrical capping of y.…”

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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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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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Entropy collapse versus entropy rigidity for Reeb and Finsler flows

Cited by 3 publications

References 66 publications

Hofer's geometry and topological entropy

Hofer's geometry and topological entropy

Braid stability and the Hofer metric

Reeb orbits that force topological entropy

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