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 Published  June 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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

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

[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.

[17]

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

[18]

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

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

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

[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.

[17]

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

[18]

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

[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.

[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.

[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.

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