On small gaps in the length spectrum

Dmitry  Dolgopyat; Dmitry Jakobson

doi:10.3934/jmd.2016.10.339

Article Contents

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

This issue Previous Article An Urysohn-type theorem under a dynamical constraint Next Article Typical dynamics of plane rational maps with equal degrees

On small gaps in the length spectrum

Dmitry Dolgopyat^1, and
Dmitry Jakobson^2,

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

Received: January 31, 2016

Revised: May 31, 2016

Published: July 2016

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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.

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Mathematics Subject Classification: Primary: 37C25, 53C22; Secondary: 20H10, 37C20, 37D20, 53D25.

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