Midwest Dynamics and Group Actions Seminar

For a compact negatively curved Riemannian manifold (\Sigma,g), the Ruelle zeta function \zeta_{\mathrm R}(\lambda) of its geodesic flow is defined for \Re\lambda\gg 1 as a convergent product over the periods T_{\gamma} of primitive closed geodesics
\displaystyle \zeta_{\mathrm R}(\lambda)=\prod_\gamma(1-e^{-\lambda T_{\gamma}})
and extends meromorphically to the entire complex plane. If \Sigma is hyperbolic (i.e. has sectional curvature -1), then the order of vanishing m_{\mathrm R}(0) of \zeta_{\mathrm R} at \lambda=0 can be expressed in terms of the Betti numbers b_j(\Sigma). In particular, Fried proved in 1986 that when \Sigma is a hyperbolic 3-manifold,
\displaystyle m_{\mathrm R}(0)=4-2b_1(\Sigma).
I will present a recent result joint with Mihajlo Cekić, Benjamin Küster, and Gabriel Paternain: when \dim\Sigma=3 and g is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely
\displaystyle m_{\mathrm R}(0)=4-b_1(\Sigma).
This is in contrast with dimension 2 where m_{\mathrm R}(0)=b_1(\Sigma)-2 for all negatively curved metrics. The proof uses the microlocal approach of expressing m_{\mathrm R}(0) as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott–Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations. Speaker(s): Semyon Dyatlov (MIT)