The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
M. Cekić, B. Delarue, S. Dyatlov, G.P. Paternain, Inventiones Mathematicae 229 (2022) 303–394.
Download
No fulltext has been uploaded.
Journal Article
| Published
| English
Author
Cekić, Mihajlo;
Delarue, BenjaminLibreCat;
Dyatlov, Semyon;
Paternain, Gabriel P.
Department
Abstract
<jats:title>Abstract</jats:title><jats:p>We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Sigma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Σ</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> with Betti number <jats:inline-formula><jats:alternatives><jats:tex-math>$$b_1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula>, the order of vanishing of the Ruelle zeta function at zero equals <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-b_1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mn>4</mml:mn>
<mml:mo>-</mml:mo>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>, while in the hyperbolic case it is equal to <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-2b_1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mn>4</mml:mn>
<mml:mo>-</mml:mo>
<mml:mn>2</mml:mn>
<mml:msub>
<mml:mi>b</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\Sigma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>Σ</mml:mi>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> with harmonic 1-forms on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Sigma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Σ</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>.</jats:p>
Keywords
Publishing Year
Journal Title
Inventiones mathematicae
Volume
229
Issue
1
Page
303-394
LibreCat-ID
Cite this
Cekić M, Delarue B, Dyatlov S, Paternain GP. The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. Inventiones mathematicae. 2022;229(1):303-394. doi:10.1007/s00222-022-01108-x
Cekić, M., Delarue, B., Dyatlov, S., & Paternain, G. P. (2022). The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. Inventiones Mathematicae, 229(1), 303–394. https://doi.org/10.1007/s00222-022-01108-x
@article{Cekić_Delarue_Dyatlov_Paternain_2022, title={The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds}, volume={229}, DOI={10.1007/s00222-022-01108-x}, number={1}, journal={Inventiones mathematicae}, publisher={Springer Science and Business Media LLC}, author={Cekić, Mihajlo and Delarue, Benjamin and Dyatlov, Semyon and Paternain, Gabriel P.}, year={2022}, pages={303–394} }
Cekić, Mihajlo, Benjamin Delarue, Semyon Dyatlov, and Gabriel P. Paternain. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” Inventiones Mathematicae 229, no. 1 (2022): 303–94. https://doi.org/10.1007/s00222-022-01108-x.
M. Cekić, B. Delarue, S. Dyatlov, and G. P. Paternain, “The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds,” Inventiones mathematicae, vol. 229, no. 1, pp. 303–394, 2022, doi: 10.1007/s00222-022-01108-x.
Cekić, Mihajlo, et al. “The Ruelle Zeta Function at Zero for Nearly Hyperbolic 3-Manifolds.” Inventiones Mathematicae, vol. 229, no. 1, Springer Science and Business Media LLC, 2022, pp. 303–94, doi:10.1007/s00222-022-01108-x.
Google Scholar