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Uspekhi Mat. Nauk, 2021, Volume 76, Issue 4(460), Pages 3–36 (Mi umn10008)  

Surveys

Chaos and integrability in $\operatorname{SL}(2,\mathbb R)$-geometry

A. V. Bolsinovabc, A. P. Veselovabd, Y. Yee

a Department of Mathematical Sciences, Loughborough University, Loughborough, UK
b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
c Moscow Center for Fundamental and Applied Mathematics
d Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
e Xi'an Jiaotong-Liverpool University, Suzhou, China

Abstract: We review the integrability of the geodesic flow on a threefold $\mathcal M^3$ admitting one of the three group geometries in Thurston's sense. We focus on the $\operatorname{SL}(2,\mathbb R)$ case. The main examples are the quotients $\mathcal M^3_\Gamma=\Gamma\backslash \operatorname{PSL}(2,\mathbb R)$, where $\Gamma \subset \operatorname{PSL}(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the corresponding phase space $T^*\mathcal M_\Gamma^3$ contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively.
As a concrete example we consider the case of the modular threefold with the modular group $\Gamma=\operatorname{PSL}(2,\mathbb Z)$. In this case $\mathcal M^3_\Gamma$ is known to be homeomorphic to the complement of a trefoil knot $\mathcal K$ in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to $\mathcal M^3_\Gamma$ produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on $\mathcal M^3_\Gamma$.
Bibliography: 60 titles.

Keywords: 3D geometries in the sense of Thurston, geodesic flows, integrability.

Funding Agency Grant Number
Russian Science Foundation 17-11-01303
20-11-20214
The work of A. V. Bolsinov and A. P. Veselov was partially supported by the Russian Science Foundation under grants no. 17-11-01303 (AVB) and 20-11-20214 (APV).


DOI: https://doi.org/10.4213/rm10008

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English version:
Russian Mathematical Surveys, 2021, 76:4, 557–586

Bibliographic databases:

UDC: 514.765+515.162.32+517.913
MSC: Primary 37D40, 37J35; Secondary 57M50
Received: 10.05.2021

Citation: A. V. Bolsinov, A. P. Veselov, Y. Ye, “Chaos and integrability in $\operatorname{SL}(2,\mathbb R)$-geometry”, Uspekhi Mat. Nauk, 76:4(460) (2021), 3–36; Russian Math. Surveys, 76:4 (2021), 557–586

Citation in format AMSBIB
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\by A.~V.~Bolsinov, A.~P.~Veselov, Y.~Ye
\paper Chaos and integrability in $\operatorname{SL}(2,\mathbb R)$-geometry
\jour Uspekhi Mat. Nauk
\yr 2021
\vol 76
\issue 4(460)
\pages 3--36
\mathnet{http://mi.mathnet.ru/umn10008}
\crossref{https://doi.org/10.4213/rm10008}
\transl
\jour Russian Math. Surveys
\yr 2021
\vol 76
\issue 4
\pages 557--586
\crossref{https://doi.org/10.1070/RM10008}
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