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Publication . Article . Preprint . 2021

On Limit Sets for Geodesics of Meromorphic Connections

Novikov, Dmitry; Shapiro, Boris; Tahar, Guillaume;
Open Access
Published: 24 Dec 2021 Journal: Journal of Dynamical and Control Systems (issn: 1079-2724, eissn: 1573-8698, Copyright policy )
Publisher: Springer Science and Business Media LLC
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 Poincar\'{e}-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.
Comment: 15 pages, 4 figures
Subjects

Control and Optimization, Numerical Analysis, Algebra and Number Theory, Control and Systems Engineering, Dynamical Systems (math.DS), Differential Geometry (math.DG), FOS: Mathematics, [2010] Primary 37F75, Secondary 32S65, Mathematics - Dynamical Systems, Mathematics - Differential Geometry

Related Organizations
12 references, page 1 of 2

[1] M. Abate, F. Bianchi, A Poincare{Bendixson theorem for meromorphic connections on Riemann surfaces, Math. Z., 282, 247{272, 2016.

[2] M. Abate, F. Tovena, Poincare{Bendixson theorem for meromorphic connections and homogeneous vector elds, J. Di erential Equations, 251, Issue 9, 2612{2684, 2011. [OpenAIRE]

[3] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Moeller, Strata of k-di erentials, Algebraic Geometry, 6, Issue 2, 196{233, 2019.

[4] A. Cheritat, and G. Tahar, Beltrami forms, imaginary curvature and the measurable Riemann mapping theorem, in preparation.

[5] E. Duryev, C. Fougeron, and S. Ghazouani, Dilation surfaces and their Veech groups, Journal of Modern Dynamics, 2019, 14, Issue 1, 121{151. [OpenAIRE]

[6] G. Frobenius, U ber das Pfa sche Problem", J. fur Reine und Angew. Math., 82 (1877) 230{ 315.

[7] R. Hotta, K. Takeuchi, T. Tanisaki, Theory of Meromorphic Connections, D-Modules, Perverse Sheaves, and Representation Theory, 127{159, Progress in Mathematics book series (PM, volume 236).

[8] D. Novikov, S. Yakovenko, Lectures on meromorphic at connections In: Y. Ilyashenko, C. Rousseau (eds), Normal Forms, Bifurcations and Finiteness problems in Di erential Equations, Kluwer 2004, 387{430.

[9] K. Rakhimov, Flat structure of meromorphic connections on Riemann surfaces, arXiv:2011.04901, 2020.

[10] B. Shapiro, and G. Tahar, On the existence of quasi-Strebel structures for meromorphic k-di erentials, arxiv: 2002.10280, submitted.

Funded by
EC| EffectiveTG
Project
EffectiveTG
Effective Methods in Tame Geometry and Applications in Arithmetic and Dynamics
  • Funder: European Commission (EC)
  • Project Code: 802107
  • Funding stream: H2020 | ERC | ERC-STG
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