The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
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222_2022_Article_1108.pdf
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Author(s)
Cekić, Mihajlo
Delarue, Benjamin
Dyatlov, Semyon
Paternain, Gabriel P.
Date Issued
March 11, 2022
Publisher
Springer Berlin Heidelberg
Citation
Cekić, Mihajlo, Delarue, Benjamin, Dyatlov, Semyon and Paternain, Gabriel P. 2022. "The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds."
Version
Final published version
Abstract
Abstract
We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold
$$\Sigma $$
Σ
with Betti number
$$b_1$$
b
1
, the order of vanishing of the Ruelle zeta function at zero equals
$$4-b_1$$
4
-
b
1
, while in the hyperbolic case it is equal to
$$4-2b_1$$
4
-
2
b
1
. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle
$$S\Sigma $$
S
Σ
with harmonic 1-forms on
$$\Sigma $$
Σ
.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00222-022-01108-x
