Journal of Modern Dynamics (JMD)

On small gaps in the length spectrum

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

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Dmitry Dolgopyat - Department of Mathematics, University of Maryland, Mathematics Building, College Park, MD 20742-4015, United States (email)
Dmitry Jakobson - Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Montréal QC H3A 2K6, Canada (email)

Abstract: 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.

Keywords:  Length spectrum, negatively curved manifolds, hyperbolicity, prevalence, Diophantine approximations.
Mathematics Subject Classification:  Primary: 37C25, 53C22; Secondary: 20H10, 37C20, 37D20, 53D25.

Received: February 2016;      Revised: June 2016;      Available Online: August 2016.