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Lower bounds for Lyapunov exponents of flat bundles on curves. (English) Zbl 1386.37036

Summary: Consider a flat bundle over a complex curve. We prove a conjecture of F. Yu [Geom. Topol. 22, No. 4, 2253–2298 (2018; Zbl 1391.14053)] that the sum of the top \(k\) Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank-\(k\) holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.
Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg for Calabi-Yau-type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)
30F60 Teichmüller theory for Riemann surfaces

Citations:

Zbl 1391.14053
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References:

[1] 10.1007/BF01393900 · Zbl 0663.30044 · doi:10.1007/BF01393900
[2] 10.4007/annals.2010.172.139 · Zbl 1203.37049 · doi:10.4007/annals.2010.172.139
[3] 10.1112/S0010437X13007550 · Zbl 1311.14010 · doi:10.1112/S0010437X13007550
[4] ; Carlson, Period mappings and period domains. Cambridge Studies in Advanced Mathematics, 85 (2003) · Zbl 1030.14004
[5] ; Deligne, Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, 163 (1970) · Zbl 0244.14004
[6] 10.1215/00127094-2017-0012 · Zbl 1381.57013 · doi:10.1215/00127094-2017-0012
[7] ; Doran, Mirror symmetry, V. AMS/IP Stud. Adv. Math., 38, 517 (2006)
[8] 10.3934/jmd.2011.5.319 · Zbl 1254.32019 · doi:10.3934/jmd.2011.5.319
[9] 10.1007/s10240-013-0060-3 · Zbl 1305.32007 · doi:10.1007/s10240-013-0060-3
[10] 10.1093/imrn/rnx044 · Zbl 1408.32017 · doi:10.1093/imrn/rnx044
[11] 10.1215/00127094-3715806 · Zbl 1370.37066 · doi:10.1215/00127094-3715806
[12] 10.2307/3062150 · Zbl 1034.37003 · doi:10.2307/3062150
[13] 10.1007/BF02392390 · Zbl 0209.25701 · doi:10.1007/BF02392390
[14] 10.1215/00127094-3165969 · Zbl 1334.22010 · doi:10.1215/00127094-3165969
[15] 10.1007/BF01404756 · Zbl 0461.14004 · doi:10.1007/BF01404756
[16] ; Kontsevich, Proceedings of the International Congress of Mathematicians, 1, 120 (1995) · Zbl 0846.53021
[17] ; Kontsevich, The mathematical beauty of physics. Adv. Ser. Math. Phys., 24, 318 (1997)
[18] 10.1007/s00222-003-0303-x · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x
[19] ; Krikorian, Séminaire Bourbaki, 2003/2004. Astérisque, 299, 59 (2005)
[20] 10.1007/BF01420526 · Zbl 0454.14006 · doi:10.1007/BF01420526
[21] ; Möller, Moduli spaces of Riemann surfaces. IAS/Park City Math. Ser., 20, 267 (2013) · Zbl 1272.30002
[22] ; Oseledets, Trudy Moskov. Mat. Obshch., 19, 179 (1968)
[23] 10.1007/BF01463870 · Zbl 0548.14004 · doi:10.1007/BF01463870
[24] 10.1007/BF02566705 · Zbl 0484.53053 · doi:10.1007/BF02566705
[25] 10.1017/CBO9780511608773 · doi:10.1017/CBO9780511608773
[26] 10.1007/BF01389674 · Zbl 0278.14003 · doi:10.1007/BF01389674
[27] 10.1090/S0002-9904-1977-14210-9 · Zbl 0354.14005 · doi:10.1090/S0002-9904-1977-14210-9
[28] 10.2307/1990994 · Zbl 0669.58008 · doi:10.2307/1990994
[29] 10.2307/1990935 · Zbl 0713.58012 · doi:10.2307/1990935
[30] 10.1215/00127094-2410655 · Zbl 1287.22005 · doi:10.1215/00127094-2410655
[31] 10.1007/978-3-663-14115-0 · doi:10.1007/978-3-663-14115-0
[32] 10.1007/978-3-322-90166-8 · Zbl 0889.33008 · doi:10.1007/978-3-322-90166-8
[33] 10.2140/gt.2018.22.2253 · Zbl 1391.14053 · doi:10.2140/gt.2018.22.2253
[34] 10.3934/jmd.2013.7.209 · Zbl 1273.32019 · doi:10.3934/jmd.2013.7.209
[35] 10.1007/978-3-540-31347-2_13 · doi:10.1007/978-3-540-31347-2_13
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