arXiv subjects

Ana Rechtman

Publications and source records attributed to Ana Rechtman.

At least 19 recordsLinked to original sources

arXiv2025

Left-handed geodesic flow of spheres of revolution

A flow on a 3-manifold is left-handed if any two ergodic invariant measures have negative linking number. We prove that on a 2-sphere of revolution whose curvature is $1/4$-pinched the geodesic flow is left-handed. Conversely, for every $\delta\le 1/4$, we construct a 2-sphere whose curvature is $\delta$-pinched and whose geodesic flow is not left-handed. This gives credit to a conjecture of Florio and Hryniewicz that $1/4$ is the optimal pinching constant for left-handedness among arbitrary positively curved 2-spheres.

math.DS↗

arXiv2025

Homoclinic orbits, Reeb chords and nice Birkhoff sections for Reeb flows in 3D

We prove that for a $C^\infty$-generic contact form defining a given co-oriented contact structure on a closed $3$-manifold, every hyperbolic periodic Reeb orbit admits a transverse homoclinic connection in each of the branches of its stable and unstable manifolds. We exploit this result to prove that for a $C^\infty$-generic contact form defining a given co-oriented contact structure, given any finite collection $\Gamma$ of periodic Reeb orbits and any Legendrian link $L$, there exists a global surface of section (embedded Birkhoff section) for the Reeb flow that contains $\Gamma$ in its boundary, and that contains in its interior a Legendrian link that is Legendrian isotopic to $L$ by a $C^0$-small isotopy. Finally we prove that if the Reeb vector field admits a $\partial$-strong Birkhoff section then every Legendrian knot has infinitely many geometrically distinct Reeb chords, except possibly when the ambient manifold is a lens space or the sphere and the Reeb flow has exactly two periodic orbits. In particular, $C^\infty$-generically on the contact form there are infinitely many geometrically distinct Reeb chords for every Legendrian knot. In the case of geodesic flows, every Legendrian knot has infinitely many disjoint chords, without any further assumptions.

math.SG↗

arXiv2025

Properties of contact toric structures and concave boundaries of linear plumbings

We consider plumbings of symplectic disk bundles over spheres admitting concave contact boundary, with the goal of understanding the geometric properties of the boundary contact structure in terms of the data of the plumbing. We focus on the linear plumbing case in this article. We study the properties of the contact structure using two different sets of tools. First, we prove that all such contact manifolds have a global contact toric structure, and use tools from toric geometry to identify when the contact structure is tight versus overtwisted. Second, we study algebraic torsion measurements from embedded contact homology (ECH) for these concavely induced contact manifolds, which has largely been unexplored. We develop a toolkit establishing existence and constraints of pseudoholomorphic curves adapted to the Morse-Bott Reeb dynamics of these plumbing examples, to provide the ECH algebraic torsion and contact invariant calculations for the concave boundaries of linear plumbings.

math.SG↗

arXiv2022

Periodic orbits and Birkhoff sections of stable Hamiltonian structures

Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on 3-manifolds: on one hand, we classify all the examples with finitely many periodic orbits under a non-degeneracy condition; on the other, we give sufficient conditions for the existence of a supporting broken book decomposition and for the existence of a Birkhoff section.

math.DS↗

arXiv2022

Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

In this paper we use broken book decompositions to study Reeb flows on closed $3$-manifolds. We show that if the Liouville measure of a nondegenerate contact form can be approximated by periodic orbits, then there is a Birkhoff section for the associated Reeb flow. In view of Irie's equidistribution theorem, this is shown to imply that the set of contact forms whose Reeb flows have a Birkhoff section contains an open and dense set in the $C^\infty$-topology. We also show that the set of contact forms whose Reeb flows have positive topological entropy is open and dense in the $C^\infty$-topology.

math.DS↗

arXiv2021

Kuperberg Dreams

The ``Seifert Conjecture'' stated, ``Every non-singular vector field on the 3-sphere ${\mathbb S}^3$ has a periodic orbit''. In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is then used to construct counter-examples to the Seifert Conjecture for smooth flows on the 3-sphere, and on compact 3-manifolds in general. The dynamics of the flows in these plugs have been extensively studied, with more precise results known in special ``generic'' cases of the construction. Moreover, the dynamical properties of smooth perturbations of Kuperberg's construction have been considered. In this work, we discuss the genesi of Kuperberg's construction as an evolution from the Schweitzer construction of an aperiodic plug. We discuss some of the known results for Kuperberg flows, and discuss some of the many interesting questions and problems that remain open about their dynamical and ergodic properties.

math.DS↗

arXiv2020

On the existence of supporting broken book decompositions for contact forms in dimension 3

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy. Moreover, we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the sphere or on a lens space.

math.DS↗

arXiv2019

Vector fields and genus in dimension 3

Given a flow on a 3-dimensional integral homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. We illustrate the formula with several examples, in particular with surfaces of low genus. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to the helicity.

math.DS↗

arXiv2019

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

math.DG↗

arXiv2016

Perspectives on Kuperberg flows

The "Seifert Conjecture" asks, "Does every non-singular vector field on the 3-sphere ${\mathbb S}^3$ have a periodic orbit?" In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is then used to construct counter-examples to the Seifert Conjecture for smooth flows on the $3$-sphere, and on compact 3-manifolds in general. The dynamics of the flows in these plugs have been extensively studied, with more precise results known in special "generic" cases of the construction. Moreover, the dynamical properties of smooth perturbations of Kuperberg's construction have been considered. In this work, we recall some of the results obtained to date for the Kuperberg flows and their perturbations. Then the main point of this work is to focus attention on how the known results for Kuperberg flows depend on the assumptions imposed on the flows, and to discuss some of the many interesting questions and problems that remain open about their dynamical and ergodic properties.

math.DS↗

arXiv2016

The trunkenness of a volume-preserving vector field

We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.

math.DS↗

arXiv2016

Equivalence of Deterministic walks on regular lattices on the plane

We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every site that can be in one of two states that force the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices, B. Z. Webb and E. G. D. Cohen, proved that given an initial position and velocity of the walker and an environment of one type of scatterers, mirrrors or rotators, there is an environment of the other type such that the walks on both environments are equivalent, meaning they visit the same sites at the same time steps. We prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices. The proofs are based both on the geometry of the lattice and the structure of the scattering rule.

nlin.CG↗

arXiv2016

Aperiodicity at the boundary of chaos

We consider the dynamical properties of $C^{\infty}$-variations of the flow on an aperiodic Kuperberg plug ${\mathbb K}$. Our main result is that there exists a smooth 1-parameter family of plugs ${\mathbb K}_{\epsilon}$ for $\epsilon \in (-a,a)$ and $a<1$, such that: (1) The plug ${\mathbb K}_0 = {\mathbb K}$ is a generic Kuperberg plug; (2) For $\epsilon <0$, the flow in the plug ${\mathbb K}_{\epsilon}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) For $\epsilon > 0$, the flow in the plug ${\mathbb K}_{\epsilon}$ has positive topological entropy, and an abundance of periodic orbits.

math.DS↗

arXiv2014

Periodic orbits of Euler vector fields on 3-manifolds

In this paper we study the existence of periodic orbits in the flow of non-singular steady Euler fields $X$ on closed 3-manifolds, that is $X$ is a solution of time independent Euler equations. We show, that when $X$ is $C^2$ the flow always posses a periodic orbit unless the manifold is a torus bundle over the circle. This result generalizes preavious result of J. Etnyre and R. Ghrist by weakening the real analytic hypothesis to $C^2$.

math.DS↗

arXiv2013

The dynamics of generic Kuperberg flows

In this work, we study the dynamical properties of Krystyna Kuperberg's aperiodic flows on $3$-manifolds. We introduce the notion of a ``zippered lamination'', and with suitable generic hypotheses, show that the unique minimal set for such a flow is an invariant zippered lamination. We obtain a precise description of the topology and dynamical properties of the minimal set, including the presence of non-zero entropy-type invariants and chaotic behavior. Moreover, we show that the minimal set does not have stable shape, yet satisfies the Mittag-Leffler condition for homology groups.

math.DS↗

arXiv2011

Averaging sequences

In the spirit of Goodman-Plante average condition for the existence of a transverse invariant measure for foliations, we give an averaging condition to find tangentially smooth measures with prescribed Radon-Nikodym cocycle. Harmonic measures are examples of tangentially smooth measures for foliations and laminations. We also present sufficient hypothesis on the averaging condition under which the tangentially smooth measure is harmonic.

math.DS↗

arXiv2011

Minimal Følner foliations are amenable

For finitely generated groups, amenability and Følner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Følner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\infty$ foliations of closed manifolds that are Følner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive answer to a question of Kaimanovich. The equivalence is stated with respect to transverse invarian measures or some tangentially smooth measures. The latter include harmonic measures, and in this case the Følner condition has to be replaced by $η$-Følner (where the usual volume is modified by the modular form $η$ of the measure).

math.DS↗

arXiv2011

The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation

Helmut Hofer introduced in '93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3--manifolds $(M,ξ)$ with $π_2(M) \ne 0$. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

math.DS↗