Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the rigidity of Zoll magnetic systems on surfaces
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.
rate research
Read More
We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily small rescalings.
In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^infty$ Riemannian metric $g$, whose genus $mathfrak{g} geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of $M$ is ergodic with respect to the Liouville measure, under the assumption that the set of points on $M$ with negative curvature has at most finitely many connected components.
In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
It was proved in cite{NS1} that obstacles $K$ in $R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
