arXiv subjects

Barney Bramham

Publications and source records attributed to Barney Bramham.

11 recordsLinked to original sources

arXiv2026

Rigidity of coarsely minimal Reeb flows

We introduce the notion of a coarsely minimal Reeb flow, generalizing the notion of minimal geodesic flow, and prove the following rigidity theorem: That a coarsely minimal Reeb flow satisfying a divergence property is orbitally equivalent to the geodesic flow of a Riemannian metric of negative sectional curvature. Without the divergence assumption, we obtain an orbital semi-equivalence. This extends a rigidity result for geodesic flows of negatively curved Riemannian metrics which is due to Gromov. We use Floer homology and Morse's hyperbolic `stability' Lemma.

math.DS↗

arXiv2024

Spectral invariants for non-compactly supported Hamiltonians on the disc, and an application to the mean action spectrum

For a symplectic isotopy on the two-dimensional disc we show that the classical spectral invariants of Viterbo [20] can be extended in a meaningful way to {\it non-compactly} supported Hamiltonians. We establish some basic properties of these extended invariants and as an application we show that Hutchings' inequality in [8] between the Calabi invariant and the mean action spectrum holds without any assumptions on the isotopy; in [8] it is assumed that the Calabi invariant is less than the rotation number (or action) on the boundary.

math.SG↗

arXiv2023

On pseudo-rotations of the annulus with generic rotation number

We show that for a Baire generic rotation number $\alpha \in \mathbb{R} / \mathbb{Z}$, the set of area preserving $C^\infty$-pseudo-rotations of the annulus $\mathbb{A}$ with rotation number $\alpha$ equals the closure of the set of area preserving $C^\infty$-pseudo-rotations which are smoothly conjugate to the rotation $R_{\alpha}$. As a corollary, a $C^\infty$-generic area preserving pseudo-rotation of the annulus with a Baire generic rotation number $\alpha$ is weakly mixing.

math.DS↗

arXiv2018

Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $π$ and equals $π$ if and only if the metric is Riemannian and Zoll.

math.SG↗

arXiv2017

Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct a sequence of tight contact forms $α_n$, $n\geq 2$, with systolic ratio arbitrarily close to $n$ and suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.

math.SG↗

arXiv2017

Contact forms with large systolic ratio in dimension three

The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume. We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio. This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena.

math.SG↗

arXiv2015

A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq π\ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.

math.DG↗

arXiv2012

Pseudo-rotations with sufficiently Liouvillean rotation number are C^0-rigid

It is an open question in smooth ergodic theory whether there exists a Hamiltonian disk map with zero topological entropy and (strong) mixing dynamics. Weak mixing has been known since Anosov and Katok first constructed examples in 1970. Currently all known examples with weak mixing are irrational pseudo-rotations with Liouvillean rotation number on the boundary. Our main result however implies that for a dense subset of Liouville numbers (strong) mixing cannot occur. Our approach involves approximating the flow of a suspension of the given disk map by pseudoholomorphic curves. Ellipticity of the Cauchy-Riemann equation allows quantitative L^2-estimates to be converted into C^0-estimates between the pseudoholomorphic curves and the trajectories of the flow on growing time scales. Arithmetic properties of the rotation number enter through these estimates.

math.DS↗

arXiv2012

Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.

math.DS↗

arXiv2011

First Steps Towards a Symplectic Dynamics

Many interesting physical systems have mathematical descriptions as finite-dimensional or infinite-dimensional Hamiltonian systems. Poincare who started the modern theory of dynamical systems and symplectic geometry developed a particular viewpoint combining geometric and dynamical systems ideas in the study of Hamiltonian systems. After Poincare the field of dynamical systems and the field of symplectic geometry developed separately. Both fields have rich theories and the time seems ripe to develop the common core with highly integrated ideas from both fields. We discuss problems which show how dynamical systems questions and symplectic ideas come together in a nontrivial way.

math.DS↗

arXiv2009

On Non-Separating Contact Hypersurfaces in Symplectic 4-Manifolds

We show that certain classes of contact 3-manifolds do not admit non-separating contact type embeddings into any closed symplectic 4-manifolds, e.g. this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic 4-manifolds. Similarly, there are symplectic 4-manifolds that can admit smoothly embedded non-separating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.

math.SG↗