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BIRS Workshop Lecture Videos

Ruelle zeta at zero for nearly hyperbolic 3-manifolds Dyatlov, Semyon


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 $$ \zeta_{\mathrm R}(\lambda)=\prod_\gamma(1-e^{-\lambda T_{\gamma}}) $$ 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, $$ m_{\mathrm R}(0)=4-2b_1(\Sigma). $$ I will present a recent result joint with Mihajlo Ceki\'c, Benjamin K\"uster, 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 $$ m_{\mathrm R}(0)=4-b_1(\Sigma). $$ 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.

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