Sub-leading asymptotics of ECH capacities
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Sub-leading asymptotics of ECH capacities Dan Cristofaro-Gardiner1 · Nikhil Savale2 Accepted: 30 August 2020 © The Author(s) 2020
Abstract In previous work (Cristofaro-Gardiner et al. in Invent Math 199:187–214, 2015), the first author and collaborators showed that the leading asymptotics of the embedded contact homology spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics. Mathematics Subject Classification 53D35 · 57R57 · 57R58
1 Introduction Asymptotic counting problems for length spectra are of significant interest in geometry, dynamics and number theory. As an example, let π (x) denote the counting function for the geodesic length spectrum, with lengths ≤ ln x, on a Riemannian manifold of negative sectional curvature. Then a classical result of Margulis [29] states E (x) := π (x) − li x h = o li x h ,
(1.1)
y as x → ∞, with li (y) := 2 lndss ∼ lnyy and h being the topological entropy of the corresponding geodesic flow. Recently, [20, Theorem 2.8] improved the error estimate above, under a pinching condition, to O li x h−α for some α > 0 with the sharp exponent being yet unknown. In the case of P S L 2 (Z) \H, a surface of constant negative curvature −1 where h = 1, there is a long history of improvements
D.C.-G. is partially supported by NSF Grant 1711976. N.S. is partially supported by the DFG funded Project CRC/TRR 191.
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Nikhil Savale [email protected] Dan Cristofaro-Gardiner [email protected]
1
Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064, USA
2
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany 0123456789().: V,-vol
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D. Cristofaro-Gardiner, N. Savale
7 35 for the error term: E (x) = O x 48 +ε [24], E (x) = O x 10 +ε [28, Theorem 1.4], 71 25 E (x) = O x 102 +ε [9] and E (x) = O x 36 +ε [38] with the conjectured E (x) = 1 O x 2 +ε expected because the Riemann hypothesis for a corresponding dynamical zeta function holds. Furthermore, counting results under homological constraints have also been proved in [1,25,27,31,32]. The aim of the present paper is to explore an analogous problem in contact dynamics corresponding to asymptotic counting of the ECH spectrum. More precisely, let Y be an oriented three-manifold. A contact form on Y is a one-form λ satisfying λ ∧ dλ > 0. A contact form determines the Reeb vector field, R, defined by λ(R) = 1, dλ(R, ·) = 0, and the contact structure ξ := Ker(λ). Closed orbits of R are called Reeb orbits. For unit cosphere bundles over Riemannian surfaces, for example, there is a canonical contact form such that Reeb orbits correspond with closed geodesics on the surface. At present, it is not even known when precisely there are infinitely many simple closed Reeb orbits on a given contact manifold [14,15]. If Y is closed and (Y , λ) is further equipped with a homology class ∈ H1 (Y ), then the embedded contact homology EC H (Y , λ, ) is defin
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