Legendrian contact homology and topological entropy

Alves, Marcelo R. R.

doi:10.1142/s1793525319500031

Cited by 12 publications

(21 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now Yomdin's theorem gives a lower bound on h top (ϕ). Note that in contrast with some other arguments of this type (see, e.g., [Al19,FS05] and references therein), the ball B can possibly have very large dimension and the map Ψ : B × L → M sending (s, x) to the image of x on L s need not be a fibration but only a submersion onto its image.…”

Section: Resultsmentioning

confidence: 99%

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Çineli¹,

Ginzburg²,

Gürel³

2021

Preprint

View full text Add to dashboard Cite

We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistence homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy.

show abstract

Section: Resultsmentioning

confidence: 99%

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Çineli¹,

Ginzburg²,

Gürel³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Remark 1.2. Positive entropy for all Reeb flows on many contact 3-manifolds different from spherizations has recently been established by Alves in [3,4,5].…”

Section: Corollarymentioning

confidence: 94%

Lower complexity bounds for positive contactomorphisms

Dahinden

2018

Isr. J. Math.

View full text Add to dashboard Cite

Let S * Q be the spherization of a closed connected manifold of dimension at least two. Consider a contactomorphism ϕ that can be reached by a contact isotopy that is everywhere positively transverse to the contact structure. In other words, ϕ is the time-1-map of a time-dependent Reeb flow. We show that the volume growth of ϕ is bounded from below by the topological complexity of the loop space of Q. Denote by ΩQ 0 (q) the component of the based loop space that contains the constant loop.Theorem. If the fundamental group of Q or the homology of ΩQ 0 (q) grows exponentially, then the volume growth of ϕ is exponential, and thus its topological entropy is positive.A similar statement holds for polynomial growths. This result generalizes work of Dinaburg, Gromov, Paternain and Petean on geodesic flows and of Macarini, Frauenfelder, Labrousse and Schlenk on Reeb flows.Our main tool is a version of Rabinowitz-Floer homology developed by Albers and Frauenfelder.

show abstract

“…• for all γ ∈ Y ⊕ and all 1-periodic orbits γ ′ of X H ⊕ representing α we have that (4), and that if the action does not depend on the capping, then ∆ H ⊕ (γ, γ ′ ) is just the positive action difference, e.g. if γ, γ ′ are contractible and…”

Section: Resultsmentioning

confidence: 99%

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Legendrian contact homology and topological entropy

Cited by 12 publications

References 44 publications

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Lower complexity bounds for positive contactomorphisms

Braid stability and the Hofer metric

Contact Info

Product

Resources

About