# On Limit Sets for Geodesics of Meromorphic Connections

@article{Novikov2021OnLS,
title={On Limit Sets for Geodesics of Meromorphic Connections},
author={Dmitry Novikov and Boris Z. Shapiro and Guillaume Tahar},
journal={Journal of Dynamical and Control Systems},
year={2021}
}
• Published 28 December 2020
• Mathematics
• Journal of Dynamical and Control Systems
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 [1, 2] 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…

## References

SHOWING 1-10 OF 15 REFERENCES

### Poincar\'e-Bendixson theorems for meromorphic connections and homogeneous vector fields

• Mathematics
• 2009
We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincare-Bendixson theorem describing recurrence properties and $\omega$-limit sets of

### Flat structure of meromorphic connections on Riemann surfaces

In the first part of the paper we study relation among meromorphic $k$-differentials, singular flat metrics and meromorphic connections. In the second part we define the notion of meromorphic

### On existence of quasi-Strebel structures for meromorphic $k$-differentials

• Mathematics
L’Enseignement Mathématique
• 2021
In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a

### Dilation surfaces and their Veech groups

• Mathematics
Journal of Modern Dynamics
• 2019
We introduce a class of objects which we call 'dilation surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples,

### Counting saddle connections in flat surfaces with poles of higher order

Flat surfaces that correspond to k-differentials on compact Riemann surfaces are of finite area provided there is no pole of order k or higher. We denote by flat surfaces with poles of higher order

### Strata of $k$-differentials

• Mathematics
Algebraic Geometry
• 2019
A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a

### Flat Surfaces

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a