\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Exponential gaps in the length spectrum

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 37C25, 53C22; Secondary: 37C20, 37D20, 53D25.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. HasselbladtIntroduction 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.
  • 加载中
SHARE

Article Metrics

HTML views(2445) PDF downloads(224) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return