Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.08262

Marked magnetic action rigidity

Louis-Brahim Beaufort·Sebastián Muñoz-Thon·Sean Richardson
View original source ↗Read source document ↗Export metadata ↓

Abstract

An exact magnetic system over a closed manifold $M$ consists of a pair $(g,α)$, where $g$ is a Riemannian metric and $α$ is a 1-form encoding a magnetic field. In this context, we consider a generalization of the marked length rigidity conjecture: does the marked magnetic action spectrum of magnetic systems with Anosov magnetic flow determine the metric and the 1-form, up to a natural obstruction? In this article we answer this question in two settings: 1) locally for systems with close metrics and 1-forms and 2) for metrics in the same conformal class.

Explore related subjects

math.DSmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Louis-Brahim Beaufort, Sebastián Muñoz-Thon, Sean Richardson. 2026-04-09. Marked magnetic action rigidity. https://arxiv.org/abs/2604.08262

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

The global attractor conjecture says that toric dynamical systems have a globally attracting point (up to linear conservation relations), or equivalently, complex balanced systems have a globally attracting point within each stoichiometric compatibility class. A proof of this conjecture implies that a large class of nonlinear dynamical systems on the positive orthant have very simple and stable dynamics. The conjecture originates from the 1972 breakthrough work by Fritz Horn and Roy Jackson, and was formulated in its current form by Horn in 1974. Toric dynamical systems can be embedded into toric differential inclusions. We show that each bounded positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the boundary of the positive orthant. We use this result to prove the global attractor conjecture. In particular, it follows that all detailed balanced mass action systems and all deficiency zero weakly reversible systems have the global attractor property.

math.DS↗
arXiv2026

Pinched Arnol'd tongues for Families of circle maps

We prove that generically for a family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} with $ϕ$ a piecewise linear forcing with $k>2$ breakpoints there is no pinching in any of its Arnol'd tongues. This is in contrast to a theorem of Campbell, Galeeva, Tresser, and Uherka who showed that with two break points there are always multiple pinchpoints in its rational tongues. We also prove that the absence of pinching is generic for Lipschitz and $C^r$ ($r>0$) forcing. The family $f_{b, ω}$ is used as a simple model for a periodically forced oscillator. The rational tongue $T_{p/q}$ represents parameter values where the system is mode-locked into a $p/q$-periodic response. The pinching of the tongues to a point represents parameter values where the system's periodic response is unstable to all perturbations in the frequency parameter $ω$. The theorems in this paper show that typically this type of instability does not occur in the families under consideration.

math.DS↗
arXiv2026

Mostly nonuniformly sectional expanding systems

We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We construct examples of a vector field of class $C^r, r \ge 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite codimension, extending the codimension two case. We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin. We also adapt the previous examples to obtain higher codimensional (i.e. with central direction of dimension greater than $2$) nonuniformly sectional expanding attractors.

math.DS↗