Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2012.14222

On limit sets for geodesics of meromorphic connections

Dmitry Novikov·Boris Shapiro·Guillaume Tahar
View original source ↗Read source document ↗Export metadata ↓

Abstract

Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behaviour of geodesics of such connections has been studied by e.g. Abate, Bianchi and Tovena in relation with generalized Poincaré-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviours of such geodesics really exist. In order to fill the gap, we use the branched affine structure induced by a Fuchsian meromorphic connection to present several examples with geodesics having infinitely many self-intersections and quite peculiar omega-limit sets.

Explore related subjects

math.DSmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dmitry Novikov, Boris Shapiro, Guillaume Tahar. 2025-08-18. On limit sets for geodesics of meromorphic connections. https://doi.org/10.1007/s10883-021-09584-9

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

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that no admissible choice of weighting function can, in general, overcome the lower bounds imposed by this regularity. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.

math.DS↗
arXiv2026

Spectral theory of frame flows on closed hyperbolic manifolds

We prove a resolvent estimate for the generator of the frame flow on hyperbolic manifolds away from vertical lines of resonances. A byproduct of the proof is an optimal essential spectral gap property for the generator, hence giving another proof of exponential mixing of frame flows with respect to the volume measure of the frame bundle. This extends the result of [https://arxiv.org/abs/2005.08387v2] in dimension 3 to any dimension. We make extensive use of the Borel-Weil calculus developed in [https://arxiv.org/abs/2405.14846] to overcome difficulties of this higher-dimensional case.

math.DS↗