# 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., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc, Providence, R. I., 1970, 1-3. [2] M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, Diophantine properties of nilpotent Lie groups, Compos. Math., 151 (2015), 1157-1188. doi: 10.1112/S0010437X14007854. [3] D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp. [4] D. Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 675-709, 896. [5] V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, W.A. Benjamin, Inc., 1968. [6] A. Baker and G. Wustholz, Logarithmic Forms and Diophantine Geometry, New Math. Monographs, 9, Cambridge University Press, Cambridge, 2007. [7] V. Bangert, Mather sets for twist maps and geodesics on tori, in Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988, 1-56. [8] L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211. [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., 470, Springer, 1975. [10] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. [11] E. Breuillard, Heights on $SL_2$ and free subgroups, in Geometry, Rigidity, and Group Actions, Chicago Lectures in Math., Univ. Chicago Press, 2011, 455-493. [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), 344-355. [13] D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4. [14] P. Eberlein, When is a geodesic flow of Anosov type? I, II, Jour. Diff. Geometry, 8 (1973), 437-463, 565-577. [15] J. Franchi and Y. Le Jan, Hyperbolic Dynamics and Brownian Motion. An Introduction, Oxford Math. Monographs. Oxford Univ. Press, Oxford, 2012. doi: 10.1093/acprof:oso/9780199654109.001.0001. [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), 51-85. doi: 10.1007/PL00011157. [17] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $\mathbfR$-rank 1 semisimple Lie groups, Ann. of Math., 92 (1970), 279-326. doi: 10.2307/1970838. [18] A. Glutsyuk, Instability of nondiscrete free subgroups in Lie groups, Transform. Groups, 16 (2011), 413-479. doi: 10.1007/s00031-011-9134-9. [19] B. Hasselblatt, Hyperbolic dynamical systems, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 239-319. doi: 10.1016/S1874-575X(02)80005-4. [20] D. Hejhal, Selberg Trace Formula for $PSL(2,\mathbbR)$, Vol. I, Lecture Notes in Math., 548, Springer, 1976. [21] D. Jakobson, I. Polterovich and J. Toth, Lower Bounds for the Remainder in Weyl's Law on Negatively Curved Surfaces, IMRN 2007, article 142. doi: 10.1093/imrn/rnm142. [22] V. Yu. Kaloshin, Growth rate of the number of periodic points, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Publ., Dordrecht, 2004, 355-385. doi: 10.1007/978-94-007-1025-2_10. [23] V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3), Geom. Funct. Anal., 11 (2001), 953-970. doi: 10.1007/s00039-001-8222-8. [24] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 137-173. [25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Math. and its Apps, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. [26] G. Knieper and H. Weiss, $C^\infty$ genericity of positive topological entropy for geodesic flows on $S^2$, J. Differential Geom., 62 (2002), 127-141. [27] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces, Comm. Math. Phys., 161 (1994), 419-432. doi: 10.1007/BF02099785. [28] G. A. Margulis, Arithmetic properties of discrete subgroups, Uspehi Mat. Nauk, 29 (1974), 49-98. [29] G. A. Margulis, Arithmeticity of nonuniform lattices in weakly noncompact groups, Funkcional. Anal. i Prilozhen, 9 (1975), 35-44. [30] G. Margulis, Discrete groups of motions of manifolds of non-positive curvature, in Proceedings of the ICM (Vancouver, B.C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 21-34. [31] G. Margulis, On some Aspects of the Theory of Anosov Systems, Translated from the Russian by V. V. Szulikowska, With a survey by R. Sharp: Periodic Orbits of Hyperbolic Flows, Springer Monographs in Math., Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1. [32] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. [33] J. Milnor, A note on the curvature and fundamental group, J. Diff. Geom., 2 (1968), 1-7. [34] G. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, IHES Publ. Math., 34 (1968), 53-104. [35] W. Parry, Equilibrium states and weighted uniform distribution of closed orbits, in Dynamical Systems (College Park, MD 1986-87), Lecture Notes in Math., 1342, Springer, 1988, 617-625. doi: 10.1007/BFb0082850. [36] W. Parry and M. Pollicott, Zeta functions and closed orbit structure for hyperbolic systems, Asterisque, 187-188 (1990), 268pp. [37] 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. [38] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory, Mathematical Research Letters, 12 (2005), 239-249. doi: 10.4310/MRL.2005.v12.n2.a9. [39] B. Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity, Proceedings AMS, 78 (1980), 455-456. doi: 10.1090/S0002-9939-1980-0553396-1. [40] D. Ruelle, Resonances for axiom A flows, J. Diff. Geom., 25 (1987), 99-116. [41] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to Function Theory (Internat. Colloq. Function Theory, Bombay, 1960), Tata Inst. of Fundamental Research, Bombay, 147-164. [42] Y. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. [43] K. A. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan, 27 (1975), 600-612. doi: 10.2969/jmsj/02740600. [44] W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1, Edited by Silvio Levy, Princeton Math. Series, 35, Princeton University Press, Princeton, NJ, 1997. [45] B. L. van der Waerden, Algebra, Vol. I, Springer, New York, 1991. doi: 10.1007/978-1-4612-4420-2. [46] P. Varju, Diophantine property in the group of affine transformation of the line, Acta Sci. Math. (Szeged), 80 (2014), 447-458. doi: 10.14232/actasm-013-757-6. [47] Y. Yomdin, Remez-type inequality for discrete sets, Israel J. Math., 186 (2011), 45-60. doi: 10.1007/s11856-011-0131-4.

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##### References:
 [1] R. Abraham, Bumpy metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc, Providence, R. I., 1970, 1-3. [2] M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, Diophantine properties of nilpotent Lie groups, Compos. Math., 151 (2015), 1157-1188. doi: 10.1112/S0010437X14007854. [3] D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209pp. [4] D. Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 675-709, 896. [5] V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, W.A. Benjamin, Inc., 1968. [6] A. Baker and G. Wustholz, Logarithmic Forms and Diophantine Geometry, New Math. Monographs, 9, Cambridge University Press, Cambridge, 2007. [7] V. Bangert, Mather sets for twist maps and geodesics on tori, in Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988, 1-56. [8] L. Barreira and J. Schmeling, Sets of "non-typical'' points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211. [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., 470, Springer, 1975. [10] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. doi: 10.1007/BF01389848. [11] E. Breuillard, Heights on $SL_2$ and free subgroups, in Geometry, Rigidity, and Group Actions, Chicago Lectures in Math., Univ. Chicago Press, 2011, 455-493. [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), 344-355. [13] D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4. [14] P. Eberlein, When is a geodesic flow of Anosov type? I, II, Jour. Diff. Geometry, 8 (1973), 437-463, 565-577. [15] J. Franchi and Y. Le Jan, Hyperbolic Dynamics and Brownian Motion. An Introduction, Oxford Math. Monographs. Oxford Univ. Press, Oxford, 2012. doi: 10.1093/acprof:oso/9780199654109.001.0001. [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), 51-85. doi: 10.1007/PL00011157. [17] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in $\mathbfR$-rank 1 semisimple Lie groups, Ann. of Math., 92 (1970), 279-326. doi: 10.2307/1970838. [18] A. Glutsyuk, Instability of nondiscrete free subgroups in Lie groups, Transform. Groups, 16 (2011), 413-479. doi: 10.1007/s00031-011-9134-9. [19] B. Hasselblatt, Hyperbolic dynamical systems, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 239-319. doi: 10.1016/S1874-575X(02)80005-4. [20] D. Hejhal, Selberg Trace Formula for $PSL(2,\mathbbR)$, Vol. I, Lecture Notes in Math., 548, Springer, 1976. [21] D. Jakobson, I. Polterovich and J. Toth, Lower Bounds for the Remainder in Weyl's Law on Negatively Curved Surfaces, IMRN 2007, article 142. doi: 10.1093/imrn/rnm142. [22] V. Yu. Kaloshin, Growth rate of the number of periodic points, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem., 137, Kluwer Acad. Publ., Dordrecht, 2004, 355-385. doi: 10.1007/978-94-007-1025-2_10. [23] V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3), Geom. Funct. Anal., 11 (2001), 953-970. doi: 10.1007/s00039-001-8222-8. [24] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 137-173. [25] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Math. and its Apps, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. [26] G. Knieper and H. Weiss, $C^\infty$ genericity of positive topological entropy for geodesic flows on $S^2$, J. Differential Geom., 62 (2002), 127-141. [27] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces, Comm. Math. Phys., 161 (1994), 419-432. doi: 10.1007/BF02099785. [28] G. A. Margulis, Arithmetic properties of discrete subgroups, Uspehi Mat. Nauk, 29 (1974), 49-98. [29] G. A. Margulis, Arithmeticity of nonuniform lattices in weakly noncompact groups, Funkcional. Anal. i Prilozhen, 9 (1975), 35-44. [30] G. Margulis, Discrete groups of motions of manifolds of non-positive curvature, in Proceedings of the ICM (Vancouver, B.C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 21-34. [31] G. Margulis, On some Aspects of the Theory of Anosov Systems, Translated from the Russian by V. V. Szulikowska, With a survey by R. Sharp: Periodic Orbits of Hyperbolic Flows, Springer Monographs in Math., Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1. [32] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. [33] J. Milnor, A note on the curvature and fundamental group, J. Diff. Geom., 2 (1968), 1-7. [34] G. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, IHES Publ. Math., 34 (1968), 53-104. [35] W. Parry, Equilibrium states and weighted uniform distribution of closed orbits, in Dynamical Systems (College Park, MD 1986-87), Lecture Notes in Math., 1342, Springer, 1988, 617-625. doi: 10.1007/BFb0082850. [36] W. Parry and M. Pollicott, Zeta functions and closed orbit structure for hyperbolic systems, Asterisque, 187-188 (1990), 268pp. [37] 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. [38] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory, Mathematical Research Letters, 12 (2005), 239-249. doi: 10.4310/MRL.2005.v12.n2.a9. [39] B. Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity, Proceedings AMS, 78 (1980), 455-456. doi: 10.1090/S0002-9939-1980-0553396-1. [40] D. Ruelle, Resonances for axiom A flows, J. Diff. Geom., 25 (1987), 99-116. [41] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, in Contributions to Function Theory (Internat. Colloq. Function Theory, Bombay, 1960), Tata Inst. of Fundamental Research, Bombay, 147-164. [42] Y. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64. [43] K. A. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan, 27 (1975), 600-612. doi: 10.2969/jmsj/02740600. [44] W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1, Edited by Silvio Levy, Princeton Math. Series, 35, Princeton University Press, Princeton, NJ, 1997. [45] B. L. van der Waerden, Algebra, Vol. I, Springer, New York, 1991. doi: 10.1007/978-1-4612-4420-2. [46] P. Varju, Diophantine property in the group of affine transformation of the line, Acta Sci. Math. (Szeged), 80 (2014), 447-458. doi: 10.14232/actasm-013-757-6. [47] Y. Yomdin, Remez-type inequality for discrete sets, Israel J. Math., 186 (2011), 45-60. doi: 10.1007/s11856-011-0131-4.
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