2020, 16: 207-223. doi: 10.3934/jmd.2020007

Exponential gaps in the length spectrum

Laboratoire d'analyse, géométrie et applications, Université Paris 13, CNRS UMR 7539, 99 avenue Jean Baptiste Clément, 93430 Villetaneuse, France

Received  November 11, 2018 Revised  March 30, 2020

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.

Citation: Emmanuel Schenck. Exponential gaps in the length spectrum. Journal of Modern Dynamics, 2020, 16: 207-223. doi: 10.3934/jmd.2020007
References:
[1]

R. Abraham, Bumpy Metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) pp. 1–3, Amer. Math. Soc., Providence, R.I, 1970., doi: 10.1090/pspum/014/0271994.  Google Scholar

[2]

N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Sci. École Norm. Sup. (4), 33 (2000), 33-56.  doi: 10.1016/S0012-9593(00)00102-6.  Google Scholar

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Y. Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compositio Math., 27 (1973), 83–106; Ibid., 27 (1973), 159–184.  Google Scholar

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J. Chazarain, Formule de Poisson pour les variétés riemanniennes, Invent. Math., 24 (1974), 65-82.  doi: 10.1007/BF01418788.  Google Scholar

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J. J. Duistermaat and V. Guillemin, The spectrum of positive eliptic operators and periodic bicharacteristics, Invent. Math., 29 (1975), 39-79.  doi: 10.1007/BF01405172.  Google Scholar

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D. Dolgopyat and D. Jakobson, On small gaps in the length spectrum, J. Mod. Dyn., 10 (2016), 339-352.  doi: 10.3934/jmd.2016.10.339.  Google Scholar

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D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390.  doi: 10.2307/121012.  Google Scholar

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P. Eberlein, When is a geodesic flow of Anosov type? I, II, J. Differential Geometry, 8 (1973), 437–463; Ibid., 8 (1973), 565–577. doi: 10.4310/jdg/1214431801.  Google Scholar

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H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank $1$ semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326.  doi: 10.2307/1970838.  Google Scholar

[10]

L. Guillopé and M. Zworski, The wave trace for riemann surfaces, Geom. Funct. Anal., 9 (1999), 1156-1168.  doi: 10.1007/s000390050110.  Google Scholar

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D. Jakobson and I. Polterovich, Estimates from below for the spectral function and for the remainder in local Weyl's law, Geom. Funct. Anal., 17 (2007), 806-838.  doi: 10.1007/s00039-007-0605-z.  Google Scholar

[12]

D. Jakobson, I. Polterovich and J. Toth, A lower bound for the remainder in weyl's law on negatively curved surfaces, Int. Math. Res. Not., (2008), Art. ID rnm142, 38 pp. doi: 10.1093/imrn/rnm142.  Google Scholar

[13]

H. Karcher, Riemannian comparison constructions, Global Differential Geometry, MAA Stud. Math., Math. Assoc. America, Washington, DC, 27 (1989), 170–222.  Google Scholar

[14] A. Katok and B. Hasselbladt, Introduction to the Theory of Modern Dynamical Systems, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[15]

G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen, 3 (1969), 89-90.   Google Scholar

[16]

G. Margulis, On some Aspects of the Theory of Anosov Systems, with a survey by R. Sharp, Springer, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar

[17]

V. Petkov, Lower bounds on the number of scattering poles for several strictly convex obstacles, Asymptot. Anal., 30 (2002), 81-91.   Google Scholar

[18]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268pp.  Google Scholar

[19]

M. Pollicott and R. Sharp, Error terms for closed orbits of hyperbolic flows, Ergodic Theory Dynam. Systems, 21 (2001), 545-562.  doi: 10.1017/S0143385701001274.  Google Scholar

[20]

V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys., 310 (2012), 675-704.  doi: 10.1007/s00220-012-1419-x.  Google Scholar

[21]

E. Schenck, Resonances near the real axis for manifolds with hyperbolic trapped sets, Amer. J. Math., 141 (2019), 757-812.  doi: 10.1353/ajm.2019.0016.  Google Scholar

show all references

References:
[1]

R. Abraham, Bumpy Metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) pp. 1–3, Amer. Math. Soc., Providence, R.I, 1970., doi: 10.1090/pspum/014/0271994.  Google Scholar

[2]

N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Sci. École Norm. Sup. (4), 33 (2000), 33-56.  doi: 10.1016/S0012-9593(00)00102-6.  Google Scholar

[3]

Y. Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compositio Math., 27 (1973), 83–106; Ibid., 27 (1973), 159–184.  Google Scholar

[4]

J. Chazarain, Formule de Poisson pour les variétés riemanniennes, Invent. Math., 24 (1974), 65-82.  doi: 10.1007/BF01418788.  Google Scholar

[5]

J. J. Duistermaat and V. Guillemin, The spectrum of positive eliptic operators and periodic bicharacteristics, Invent. Math., 29 (1975), 39-79.  doi: 10.1007/BF01405172.  Google Scholar

[6]

D. Dolgopyat and D. Jakobson, On small gaps in the length spectrum, J. Mod. Dyn., 10 (2016), 339-352.  doi: 10.3934/jmd.2016.10.339.  Google Scholar

[7]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390.  doi: 10.2307/121012.  Google Scholar

[8]

P. Eberlein, When is a geodesic flow of Anosov type? I, II, J. Differential Geometry, 8 (1973), 437–463; Ibid., 8 (1973), 565–577. doi: 10.4310/jdg/1214431801.  Google Scholar

[9]

H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank $1$ semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326.  doi: 10.2307/1970838.  Google Scholar

[10]

L. Guillopé and M. Zworski, The wave trace for riemann surfaces, Geom. Funct. Anal., 9 (1999), 1156-1168.  doi: 10.1007/s000390050110.  Google Scholar

[11]

D. Jakobson and I. Polterovich, Estimates from below for the spectral function and for the remainder in local Weyl's law, Geom. Funct. Anal., 17 (2007), 806-838.  doi: 10.1007/s00039-007-0605-z.  Google Scholar

[12]

D. Jakobson, I. Polterovich and J. Toth, A lower bound for the remainder in weyl's law on negatively curved surfaces, Int. Math. Res. Not., (2008), Art. ID rnm142, 38 pp. doi: 10.1093/imrn/rnm142.  Google Scholar

[13]

H. Karcher, Riemannian comparison constructions, Global Differential Geometry, MAA Stud. Math., Math. Assoc. America, Washington, DC, 27 (1989), 170–222.  Google Scholar

[14] A. Katok and B. Hasselbladt, Introduction to the Theory of Modern Dynamical Systems, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[15]

G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen, 3 (1969), 89-90.   Google Scholar

[16]

G. Margulis, On some Aspects of the Theory of Anosov Systems, with a survey by R. Sharp, Springer, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar

[17]

V. Petkov, Lower bounds on the number of scattering poles for several strictly convex obstacles, Asymptot. Anal., 30 (2002), 81-91.   Google Scholar

[18]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268pp.  Google Scholar

[19]

M. Pollicott and R. Sharp, Error terms for closed orbits of hyperbolic flows, Ergodic Theory Dynam. Systems, 21 (2001), 545-562.  doi: 10.1017/S0143385701001274.  Google Scholar

[20]

V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys., 310 (2012), 675-704.  doi: 10.1007/s00220-012-1419-x.  Google Scholar

[21]

E. Schenck, Resonances near the real axis for manifolds with hyperbolic trapped sets, Amer. J. Math., 141 (2019), 757-812.  doi: 10.1353/ajm.2019.0016.  Google Scholar

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