# American Institute of Mathematical Sciences

2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339

## On small gaps in the length spectrum

 1 Department of Mathematics, University of Maryland, Mathematics Building, College Park, MD 20742-4015, United States 2 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6

Received  February 2016 Revised  June 2016 Published  August 2016

We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrarily small gaps is topologically generic: this is established both for surfaces of constant negative curvature (Theorem 3.1) and for the space of negatively curved metrics (Theorem 4.1). While arbitrarily small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. One result in this direction is presented in Section 5.
Citation: Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339
##### References:
 [1] R. Abraham, Bumpy metrics,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 1. Google Scholar [2] M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, Diophantine properties of nilpotent Lie groups,, Compos. Math., 151 (2015), 1157. doi: 10.1112/S0010437X14007854. Google Scholar [3] D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967). Google Scholar [4] D. Anosov, Generic properties of closed geodesics,, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 675. Google Scholar [5] V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics,, W.A. Benjamin, (1968). Google Scholar [6] A. Baker and G. Wustholz, Logarithmic Forms and Diophantine Geometry,, New Math. Monographs, (2007). Google Scholar [7] V. Bangert, Mather sets for twist maps and geodesics on tori,, in Dynamics Reported, (1988), 1. Google Scholar [8] L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211. Google Scholar [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Math., (1975). Google Scholar [10] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. doi: 10.1007/BF01389848. Google Scholar [11] E. Breuillard, Heights on $SL_2$ and free subgroups,, in Geometry, (2011), 455. Google Scholar [12] Yu A. Brudnyi and M. I. Ganzburg, A certain extremal problem for polynomials in $n$ variables, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 344. Google Scholar [13] D. Dolgopyat, Bounded orbits of Anosov flows,, Duke Math. J., 87 (1997), 87. doi: 10.1215/S0012-7094-97-08704-4. Google Scholar [14] P. Eberlein, When is a geodesic flow of Anosov type? I, II,, Jour. Diff. Geometry, 8 (1973), 437. Google Scholar [15] J. Franchi and Y. Le Jan, Hyperbolic Dynamics and Brownian Motion. An Introduction,, Oxford Math. Monographs. Oxford Univ. Press, (2012). doi: 10.1093/acprof:oso/9780199654109.001.0001. Google Scholar [16] A. Gamburd, D. Jakobson and P. Sarnak, Spectra of elements in the group ring of $SU(2)$,, Jour. of European Math. Soc., 1 (1999), 51. doi: 10.1007/PL00011157. Google Scholar [17] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $\mathbfR$-rank 1 semisimple Lie groups,, Ann. of Math., 92 (1970), 279. doi: 10.2307/1970838. Google Scholar [18] A. Glutsyuk, Instability of nondiscrete free subgroups in Lie groups,, Transform. Groups, 16 (2011), 413. doi: 10.1007/s00031-011-9134-9. Google Scholar [19] B. Hasselblatt, Hyperbolic dynamical systems,, in Handbook of Dynamical Systems, (2002), 239. doi: 10.1016/S1874-575X(02)80005-4. Google Scholar [20] D. Hejhal, Selberg Trace Formula for $PSL(2,\mathbbR)$,, Vol. I, (1976). Google Scholar [21] D. Jakobson, I. Polterovich and J. Toth, Lower Bounds for the Remainder in Weyl's Law on Negatively Curved Surfaces,, IMRN 2007, (2007). doi: 10.1093/imrn/rnm142. Google Scholar [22] V. Yu. Kaloshin, Growth rate of the number of periodic points,, in Normal Forms, (2004), 355. doi: 10.1007/978-94-007-1025-2_10. Google Scholar [23] V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3),, Geom. Funct. Anal., 11 (2001), 953. doi: 10.1007/s00039-001-8222-8. Google Scholar [24] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 137. Google Scholar [25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Math. and its Apps, (1995). doi: 10.1017/CBO9780511809187. Google Scholar [26] G. Knieper and H. Weiss, $C^\infty$ genericity of positive topological entropy for geodesic flows on $S^2$,, J. Differential Geom., 62 (2002), 127. Google Scholar [27] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces,, Comm. Math. Phys., 161 (1994), 419. doi: 10.1007/BF02099785. Google Scholar [28] G. A. Margulis, Arithmetic properties of discrete subgroups,, Uspehi Mat. Nauk, 29 (1974), 49. Google Scholar [29] G. A. Margulis, Arithmeticity of nonuniform lattices in weakly noncompact groups,, Funkcional. Anal. i Prilozhen, 9 (1975), 35. Google Scholar [30] G. Margulis, Discrete groups of motions of manifolds of non-positive curvature,, in Proceedings of the ICM (Vancouver, (1974), 21. Google Scholar [31] G. Margulis, On some Aspects of the Theory of Anosov Systems,, Translated from the Russian by V. V. Szulikowska, (2004). doi: 10.1007/978-3-662-09070-1. Google Scholar [32] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials,, Chapman & Hall/CRC, (2003). Google Scholar [33] J. Milnor, A note on the curvature and fundamental group,, J. Diff. Geom., 2 (1968), 1. Google Scholar [34] G. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms,, IHES Publ. Math., 34 (1968), 53. Google Scholar [35] W. Parry, Equilibrium states and weighted uniform distribution of closed orbits,, in Dynamical Systems (College Park, (1342), 1986. doi: 10.1007/BFb0082850. Google Scholar [36] W. Parry and M. Pollicott, Zeta functions and closed orbit structure for hyperbolic systems,, Asterisque, 187-188 (1990), 187. Google Scholar [37] V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals,, Comm. Math. Phys., 310 (2012), 675. doi: 10.1007/s00220-012-1419-x. Google Scholar [38] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory,, Mathematical Research Letters, 12 (2005), 239. doi: 10.4310/MRL.2005.v12.n2.a9. Google Scholar [39] B. Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity,, Proceedings AMS, 78 (1980), 455. doi: 10.1090/S0002-9939-1980-0553396-1. Google Scholar [40] D. Ruelle, Resonances for axiom A flows,, J. Diff. Geom., 25 (1987), 99. Google Scholar [41] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces,, in Contributions to Function Theory (Internat. Colloq. Function Theory, (1960), 147. Google Scholar [42] Y. Sinai, Gibbs measures in ergodic theory,, Russian Math. Surveys, 27 (1972), 21. Google Scholar [43] K. A. Takeuchi, A characterization of arithmetic Fuchsian groups,, J. Math. Soc. Japan, 27 (1975), 600. doi: 10.2969/jmsj/02740600. Google Scholar [44] W. P. Thurston, Three-dimensional Geometry and Topology,, Vol. 1, (1997). Google Scholar [45] B. L. van der Waerden, Algebra,, Vol. I, (1991). doi: 10.1007/978-1-4612-4420-2. Google Scholar [46] P. Varju, Diophantine property in the group of affine transformation of the line,, Acta Sci. Math. (Szeged), 80 (2014), 447. doi: 10.14232/actasm-013-757-6. Google Scholar [47] Y. Yomdin, Remez-type inequality for discrete sets,, Israel J. Math., 186 (2011), 45. doi: 10.1007/s11856-011-0131-4. Google Scholar

show all references

##### References:
 [1] R. Abraham, Bumpy metrics,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 1. Google Scholar [2] M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, Diophantine properties of nilpotent Lie groups,, Compos. Math., 151 (2015), 1157. doi: 10.1112/S0010437X14007854. Google Scholar [3] D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967). Google Scholar [4] D. Anosov, Generic properties of closed geodesics,, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 675. Google Scholar [5] V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics,, W.A. Benjamin, (1968). Google Scholar [6] A. Baker and G. Wustholz, Logarithmic Forms and Diophantine Geometry,, New Math. Monographs, (2007). Google Scholar [7] V. Bangert, Mather sets for twist maps and geodesics on tori,, in Dynamics Reported, (1988), 1. Google Scholar [8] L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211. Google Scholar [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Math., (1975). Google Scholar [10] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. doi: 10.1007/BF01389848. Google Scholar [11] E. Breuillard, Heights on $SL_2$ and free subgroups,, in Geometry, (2011), 455. Google Scholar [12] Yu A. Brudnyi and M. I. Ganzburg, A certain extremal problem for polynomials in $n$ variables, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 344. Google Scholar [13] D. Dolgopyat, Bounded orbits of Anosov flows,, Duke Math. J., 87 (1997), 87. doi: 10.1215/S0012-7094-97-08704-4. Google Scholar [14] P. Eberlein, When is a geodesic flow of Anosov type? I, II,, Jour. Diff. Geometry, 8 (1973), 437. Google Scholar [15] J. Franchi and Y. Le Jan, Hyperbolic Dynamics and Brownian Motion. An Introduction,, Oxford Math. Monographs. Oxford Univ. Press, (2012). doi: 10.1093/acprof:oso/9780199654109.001.0001. Google Scholar [16] A. Gamburd, D. Jakobson and P. Sarnak, Spectra of elements in the group ring of $SU(2)$,, Jour. of European Math. Soc., 1 (1999), 51. doi: 10.1007/PL00011157. Google Scholar [17] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $\mathbfR$-rank 1 semisimple Lie groups,, Ann. of Math., 92 (1970), 279. doi: 10.2307/1970838. Google Scholar [18] A. Glutsyuk, Instability of nondiscrete free subgroups in Lie groups,, Transform. Groups, 16 (2011), 413. doi: 10.1007/s00031-011-9134-9. Google Scholar [19] B. Hasselblatt, Hyperbolic dynamical systems,, in Handbook of Dynamical Systems, (2002), 239. doi: 10.1016/S1874-575X(02)80005-4. Google Scholar [20] D. Hejhal, Selberg Trace Formula for $PSL(2,\mathbbR)$,, Vol. I, (1976). Google Scholar [21] D. Jakobson, I. Polterovich and J. Toth, Lower Bounds for the Remainder in Weyl's Law on Negatively Curved Surfaces,, IMRN 2007, (2007). doi: 10.1093/imrn/rnm142. Google Scholar [22] V. Yu. Kaloshin, Growth rate of the number of periodic points,, in Normal Forms, (2004), 355. doi: 10.1007/978-94-007-1025-2_10. Google Scholar [23] V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3),, Geom. Funct. Anal., 11 (2001), 953. doi: 10.1007/s00039-001-8222-8. Google Scholar [24] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 137. Google Scholar [25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Math. and its Apps, (1995). doi: 10.1017/CBO9780511809187. Google Scholar [26] G. Knieper and H. Weiss, $C^\infty$ genericity of positive topological entropy for geodesic flows on $S^2$,, J. Differential Geom., 62 (2002), 127. Google Scholar [27] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces,, Comm. Math. Phys., 161 (1994), 419. doi: 10.1007/BF02099785. Google Scholar [28] G. A. Margulis, Arithmetic properties of discrete subgroups,, Uspehi Mat. Nauk, 29 (1974), 49. Google Scholar [29] G. A. Margulis, Arithmeticity of nonuniform lattices in weakly noncompact groups,, Funkcional. Anal. i Prilozhen, 9 (1975), 35. Google Scholar [30] G. Margulis, Discrete groups of motions of manifolds of non-positive curvature,, in Proceedings of the ICM (Vancouver, (1974), 21. Google Scholar [31] G. Margulis, On some Aspects of the Theory of Anosov Systems,, Translated from the Russian by V. V. Szulikowska, (2004). doi: 10.1007/978-3-662-09070-1. Google Scholar [32] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials,, Chapman & Hall/CRC, (2003). Google Scholar [33] J. Milnor, A note on the curvature and fundamental group,, J. Diff. Geom., 2 (1968), 1. Google Scholar [34] G. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms,, IHES Publ. Math., 34 (1968), 53. Google Scholar [35] W. Parry, Equilibrium states and weighted uniform distribution of closed orbits,, in Dynamical Systems (College Park, (1342), 1986. doi: 10.1007/BFb0082850. Google Scholar [36] W. Parry and M. Pollicott, Zeta functions and closed orbit structure for hyperbolic systems,, Asterisque, 187-188 (1990), 187. Google Scholar [37] V. Petkov and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals,, Comm. Math. Phys., 310 (2012), 675. doi: 10.1007/s00220-012-1419-x. Google Scholar [38] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory,, Mathematical Research Letters, 12 (2005), 239. doi: 10.4310/MRL.2005.v12.n2.a9. Google Scholar [39] B. Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity,, Proceedings AMS, 78 (1980), 455. doi: 10.1090/S0002-9939-1980-0553396-1. Google Scholar [40] D. Ruelle, Resonances for axiom A flows,, J. Diff. Geom., 25 (1987), 99. Google Scholar [41] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces,, in Contributions to Function Theory (Internat. Colloq. Function Theory, (1960), 147. Google Scholar [42] Y. Sinai, Gibbs measures in ergodic theory,, Russian Math. Surveys, 27 (1972), 21. Google Scholar [43] K. A. Takeuchi, A characterization of arithmetic Fuchsian groups,, J. Math. Soc. Japan, 27 (1975), 600. doi: 10.2969/jmsj/02740600. Google Scholar [44] W. P. Thurston, Three-dimensional Geometry and Topology,, Vol. 1, (1997). Google Scholar [45] B. L. van der Waerden, Algebra,, Vol. I, (1991). doi: 10.1007/978-1-4612-4420-2. Google Scholar [46] P. Varju, Diophantine property in the group of affine transformation of the line,, Acta Sci. Math. (Szeged), 80 (2014), 447. doi: 10.14232/actasm-013-757-6. Google Scholar [47] Y. Yomdin, Remez-type inequality for discrete sets,, Israel J. Math., 186 (2011), 45. doi: 10.1007/s11856-011-0131-4. Google Scholar
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